Daughter  or  William 

Stuart  Smith, U.S.  Navy 


AJ 


'MO/i 


J^?. 


ELEMENTS 


op  .'»,  » 

TO,5      ' 


>      :>    >  > 


ANALYTICAL  MECHANICS. 


BY 


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W.   H.   C.   BARTLETT,   LL.D., 

COLONEL  U.  S.  A.,  RETIRED,  AND  LATE  PROFESSOR  OP  NATURAL  AND  EXPERIMENTAL  PHIL- 
OSOPHY IN  THE  U.  8.  MILITARY  ACADEMY  AT  WEST  POINT,  AND  AUTHOR  OP  "  ELE- 
MENTS OF  SYNTHETICAL  MECHANICS,  ASTRONOMY,  ACOUSTICS  AND  OPTICS." 


X 


NINTH   EDITION,  EEVISED   AND   COREECTED. 


A.   S.    BARNES    AND    COMPANY, 

NEW   YORK  AND  CHICAGO. 


Valuable  Works  to  Lealino;  Authors      v 


TN  THE 


HIGHER    MATHEMATICS. 


W.  H.  C.  BARTLETT,  L.L.D., 

?*rof.  of  jVat.  &  JZzp.   iPhilos.  in  the  U.   S.  Military  Academy,   West  Point 

\  c  <(c  <c<{  <<  IBARTLETT'S    SYNTHETIC    MECHANICS. 

,'   ,' ,r '  /c^    Elements  of  M^chank-s,  embracing  Mathematical  formulae  for  observing  and  calculating 
the  action  of  Forces  upon  Bodies — the  source  of  all  physical  phenomena. 

BARTLBTTS    ANALYTICAL    MECHANICS. 
Fer  more  advanced  students  than  the  preceding,  the  subjects  being  discussed  Analytically, 
•by  the  aid  of  Calculus. 

BARTLETT'S    ACOUSTICS    AND    OPTICS. 

Treating  Sound  and  Light  as  disturbances  of  the  normal  Equilibrium  of  an  analogous  char- 
acter, and  to  be  considered  under  the  same  general  laws. 

BARTLETT'S    ASTRONOMY. 

Spherical  Astronomy  in  its  relations  to  Celestial  Mechanics,  with  full  applications  to  lh« 
current  wants  of  Navigation,  Geography,  and  Chronology. 


A.  E.  CHURCH,  L.L.D., 

'Prof,  Mathematics  in  the   United  States  Military  Academy ,    West  jPoint. 

CHURCH'S    ANALYTICAL    GEOMETRY. 
Elements  of  Analytical  Geometry,  preserving  the  true  spirit  of  Analysis,  and  rendering  thw 
whole  subject  attractive  and  easily  acquired. 

CHURCH'S    CALCULUS. 
Elements  of  the  Differential  and  Integral  Calculus,  with  the  Calculus  of  Variations. 

CHURCH'S    DESCRIPTIVE    GEOMETRY. 

Elements  of  Descriptive  Geometry,  with  its  applications  to  Spherical  Projections,  Shades 
And  Shadows,  Perspective  and  Isometric  Projections.    2  vols. ;  Text  and  Plates  respectively. 


EDWARD   M.  COURTENAY,  LL.D., 

Z,ale  Pro  ft  Mathematics  in  the  University  of  Virginia. 

COURTENAYS    CALCULUS. 
A  treatise  on  the  Differential  and  Integral  Calculus,  and  on  the  Calculus  of  Variations. 


CHAS.  W.  HACKLEY,  S.T.D., 

Late  iProf.  of  Mathematics  and  Astronomy  in  Columbia  College. 

HACKLEY'S    TRIG-OX  OMETRY. 

A  treatise  on  Trigonometry,  Plane  and  Spherical,  with  its  application  to  Navigation  and 
Surveying,  Nautical  and  Practical  Astronomy  and  Geodesy,  with  Logarithmic,  Trigonomet- 
rical, and:  Nautical  Tables. ^^^^^^^ 

DAVIES   &   PECK, 

Department  of  Mathematics ,   Columbia  College. 

MATHEMATICAL    DICTIONARY 
And  Cyclopedia  of  Mathematical  Science,  comprising  Definitions  of  all  the  terms  employed 
in  Mathematics— an  analysis  of  each  branch,  and  of  the  whole  as  forming  a  single  science. 

C H ARLES   DAVIES,  L L. D., 

JLate  of  the   United  States  Military  Academy  and  of  Columbia  College. 

A    COMPLETK    COURSE    IN    MATHEMATICS. 

See  A.  S.  Barnes  &  Co.'s  Descriptive  Catalogue. 

— — — — —     -  . B 1 — . —  " 

Entered,  according  to  Act  of  Congress,  Li  the  year  1874,  by 

W.  H.  C.  BARTLETT. 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  :<>r  the  S01.tr  em  District  of  New  York. 


B'S   AX  AT.    MECTI.        Qjpj  QJT 


V 


TO 

« 

COLONEL  SYLVANUS  THAYER, 

OF  THE  CORPS  OF  ENGINEERS,  AND  LATE  SUPERINTENDENT  OF  THK 

UNITED  STATES  MILITARY   ACADEMY, 

ftfcij  Mark. 

18   MOST   RESPECTFULLY  AND   AFFECTIONATELY  DEDICATED 

IN    GRATITUDE    FOR    THE    PRIVILEGES 

ITS    AUTHOR    HAS    ENJOYED    UNDER    A    SYSTEM    OF    INSTRUCTION 

AND    GOVERNMENT    WHICH    GAVE    VITALITY    TO 

THE  ACADEMY, 

AND    OF    WHICH    HE    IS    THE    FATHES. 

* 


863899 


PREFACE  TO  THE  SECOND  EDITION. 


.It  is  now  six  years  since  the  publication  of  the  first  edi- 
tion of  the  present  work.  During  this  interval,  it  has  been 
corrected  and  amended  according  to  the  suggestions  of  daily 
experience  in  its  use  as  a  text-book.  It  now  appears  with  an 
additional  part,  under  the  head,  Mechanics  of  Molecules  ;  and 
this  completes — in  so  far  as  he  may  have  succeeded  in  its  ex- 
ecution— the  design  of  the  author  to  give  to  the  classes  com- 
mitted to  his  instruction,  in  the  Military  Academy,  what  has 
appeared  to  him  a  proper  elementary  basis  for  a  systematic 
study  of  the  laws  of  matter.  The  subject  is  the  action  of 
forces  upon  bodies, — the  source  of  all  physical  phenomena — and 
of  which  the  sole  and  sufficient  foundation  is  the  comprehensive 
fact,  that  all  action  is  ever  accompanied  by  an  equal,  contrary, 
and  simultaneous  reaction.  Neither  can  have  precedence  of 
the  other  in  the  order  of  time,  and  from  this  comes  that  char- 
acter of  permanence,  in  the  midst  of  endless  variety,  apparent 
in  the  order  of  nature.  A  mathematical  formula  which  shall 
express  the  laws  of  this  antagonism  will  contain  the  whole  sub- 
ject; and  whatever  of  specialty  may  mark  our  perceptions  of 
a  particular  instance,  will  be  found  to  have  its  origin  in  corre- 
sponding peculiarities  of  physical  condition,  distance,  place, 
and  time,  which  are  the  elements  of  this  formula.  Its  discus- 
sion constitutes  the  study  of  Mechanics.  All  phenomena  in 
which  bodies  have  a  part  are  its  legitimate  subjects,  and  no 
form  of  matter  under  extraneous  influences  is  exempt  from  its 


PREFACE. 

scrutiny.  It  embraces  alike,  in  their  reciprocal  action,  the 
gigantic  and  distant  orbs  of  the  celestial  regions,  and  the 
proximate  atoms  of  the  ethereal  atmosphere  which  pervades 
all  space  and  establishes  an  unbroken  continuity  upon  which 
its  Divine  Architect  and  Author  may  impress  the  power  of 
His  will  at  a  single  point  and  be  felt  everywhere.  Astronomy, 
terrestrial  physics,  and  chemistry  are  but  its  specialties ; .  it 
classifies  all  of  human  knowledge  that  relates  to  inert  matter 
into  groups  of  phenomena,  of  which  the  rationale  is  in  a  com- 
mon principle;  and  in  the  hands  of  those  gifted  with  the 
priceless  boon  of  a  copious  mathematics,  it  is  a  key  to  exter- 
nal nature. 

The  order  of  treatment  is  indicated  by  the  heads  of  Me- 
chanics of  Solids,  of  Fluids,  and  of  Molecules, — an  order  sug- 
gested by  differences  of  physical  constitution. 

The  author  would  acknowledge  his  obligation  to  the  works 
of  many  eminent  writers,  and  particularly  to  those  of  MM.  La- 
grange, Poisson,  Poncelet,  Fresnel,  Lame,  Sir  William  R.  Hamilton, 
the  Rev.  Baden  Powell,  Mr.  Airy,  Mr.  Pratt,  and  Mr.  A.  Smith. 

West:  Point,  1858 


PREFACE  TO  THE  NINTH  EDITION. 


Twenty  years  ago,  the  course  of  Mechanics  taught,  for  several 
previous  years,  to  classes  in  the  United  States  Military  Academy, 
was  published  in  the  first  edition  of  this  work. 

In  that  edition  the  following  assertion  was  made: 

"All  physical  phenomena  are  but  the  necessary  results  of  a 
perpetual  conflict  of  equal  and  opposing  forces,  and  the  mathe- 
matical formula  expressive  of  the  laws  of  this  conflict  must  involve 
the  whole  doctrine  of  Mechanics.  The  study  of  Mechanics  should, 
therefore,  be  made  to  consist  simply  in  the  discussion  of  this  for- 
mula, and  in  it  should  be  sought  the  explanation  of  all  effects 
that  arise  from  the  action  of  forces." 

From  the  single  fundamental  formula  thus  referred  to,  the  whole 
of  Analytical  Mechanics  was  then  deduced. 

That  formula  was  no  other  than  the  simple  analytical  expres- 
sion of  what  is  now  generally  called  the  law  of  the  conservation 
of  energy,  which  has  since  revolutionized  physical  science  in  nearly 
all  its  branches,  and  which  at  that  time  was  but  little  developed 
or  accepted. 

It  is  believed  that  this  not  only  was  the  first,  but  that  it  even 
still  is  the  only  treatise  on  Analytical  Mechanics  in  which  all  the 
phenomena  are  presented  as  mere  consequences  of  that  single  law. 


iv  PREFACE. 

And  in  offering  to  the  public  this  new  edition,  which  has  been 
most  carefully  revised  and  in  many  parts  rewritten,  one  of  the 
principal  objects  sought  has  been  to  render  it  more  worthy  of  use, 
by  making  it  what  it  ought  to  be  in  view  of  the  great  progress 
achieved  during  the  last  twenty-five  years,  in  consequence  chiefly 
of  the  more  general  recognition  and  acceptance  of  the  grand  law 
of  work  and  energy,  by  Newton  called  that  of  action  and  reaction. 

To  Professor  EL  S.  McCulloch  my  acknowledgments  are  due  not 
only  for  suggestions,  but  also  for  valuable  aid  in  preparing  the 
present  edition  for  the  press.  And  to  Professor  P.  S.  Michie,  my 
former  pupil  and  now  able  successor  in  the  Military  Academy  at 
West  Point,  I  am  also  much  indebted. 

Yonkers,  N.  Y.,  1874. 


CONTENTS. 


INTRODUCTION. 

PAGE 

Preliminary  Definitions.' 13 

Rest,  Motion,  Force 14 

Constitution  of  Bodies 14 

Inertia 16 

Mass 17 

Mechanics 18 

PART    I. 

Force  and  Motion 19 

Motion  and  Rest 25 

Work 26 

Varied  Motion 30 

Equilibrium 34 

General  Laws  of  Work  and  Energy 35 

Principle  of  D'Alembert 36 

Virtual  Velocities 37 

Interpretation  of  Equation  (A). 39 

Reference  to  Co  ordinate  Axes. 44 

Composition  and  Resolution  of  Oblique  Forces 50 

Parallelogram  of  Forces 53 

Parallelopipedon  of  Forces 57 

Parallel  Forces 63 

Work  of  Resultant  and  Components 72 

Work  of  Rotation 73 

Moments 76 

Composition  and  Resolution  of  Moments 78 

Translation  of  Equations  B  and  C 81 

Centre  of  Gravity . .  83 

Centre  of  Gravity  of  Lines 87 

Centre  of  Gravity  of  Surfaces 92 

Centre  of  Gravity  of  Volumes 99 

Centrobaryc  Method 104 

Centre  of  Inertia 106 

Motion  of  the  Centre  of  Inertia 107 

Rotation  around  the  Centre  of  Inertia 108 

Motion  of  Translation 110 

General  Theorem  of  Work,  Energy,  &c 110 

Stable  and  Unstable  Equilibrium 113 

Potential  Function 116 


vi  CONTENTS. 

PAGE 

Conservation  of  Energy 117 

Discussion  of  Function  II. ^ 118 

Initial  Conditions,  Direct  and  Inverse  Problems 122 

Vertical  Motion  of  Heavy  Bodies 123 

Projectiles 131 

Rotary  Motion  . .    147 

Moment  of  Inertia,  Radius  and  Centre  of  Gyration 1 59 

Motion  of  a  Body  under  Impulsion 171 

Motion  of  the  Centre  of  Inertia 171 

Motion  about  the  Centre  of  Inertia 173 

Angular  Velocity 174 

Axis  of  Instantaneous  Rotation : 175 

Axis  of  Spontaneous  Rotation 170 

Stable  and  Unstable  Rotation 177 

Motion  of  a  System  of  Bodies . 179 

Motion  of  the  Centre  of  Inertia  of  the  System 180 

Motion  of  the  System  about  its  Common  Centre  of  Inertia 181 

Conservation  of  the  Motion  of  the  Centre  of  Inertia 182 

Conservation  of  Areas 183 

Invariable  Plane 185 

Conservation  of   Kinetic  Energy 185 

Principle  of  Least  Action 187 

Planetary  Motions s. 198 

Laws  of  Central  Forces 200 

The  Orbit 206 

System  of  the  World 208 

Consequences  of  Kepler's  Laws 208 

Perturbations 213 

Coexistence  and  Superposition  of  Small  Motions  215 

Universal  Gravitation 216 

Impact  of  Bodies 221 

Constrained  Motion 228 

Constrained  Motion  on  a  Curve   and   Surfaces 230 

Constrained  Motion  about  a  Fixed   Point 255 

Constrained  Motion  about  a  Fixed   Axis 257 

Compound   Pendulum 259 

Ballistic  Pendulum 269 

Gun  Pendulum 271 

PAET    II. 

MECHANICS    OF    FLUIDS. 

Introductory  Remarks 273 

Mariotte's  Law 275 

Law  of  Pressure.  Density,  and  Temperature  275 

Equal  Transmission  of  Pressure 278 

Motion  of  Fluid  Particles 280 

Equilibrium  of  Fluids 290 

Pressure  of  Heavy  Fluids 299 


CONTENTS.  vii 

PAGK 

Equilibrium  and  Stability  of  Floating  Bodies 307 

Specific  Gravity 316 

Atmospheric  Pressure 320 

Barometer 321 

Motion  of  Heavy  Incompressible  Fluids  in   Vessels 330 

Steady  Flow   of   Fluids 342 

Steady  Motion  of  Elastic  Fluids 352 

Digression  on  the  Action  of  Heat  upon  Air 356 

New  Equations  of  Steady  Flow 359 

PART    III. 

MECHANICS    OF    MOLECULES. 

Introductory  Remarks 355 

Periodicity  of  Molecular  Condition 365 

Waves 372 

Wave  Function 373 

Wave  Velocity 380 

Relation  of  Wave  Velocity  to  Wave  Length 383 

Surface  of  Elasticity 385 

Wave  Surface 387 

Double  Wave  Velocity 392 

Umbilic  Points 395 

Molecular  Orbits 398 

Reflexion  and  Refraction 401 

Resolution  of  Living  Force  by  Deviating   Surfaces 404 

Polarization  by  Reflexion  and  Refraction 408 

Diffusion  and  Decay  of  Living  Force  414 

Interference 415 

Inflexion 420 

PAET    IV. 

APPLICATIONS  TO   SIMPLE  MACHINES,  PUMPS,  &c. 

General  Principles  of  all  Machines 435 

Friction 427 

Stiffness  of  Cordage 435 

Friction  on  Pivots 440 

Friction  on  Trunnions 445 

The  Cord  as  a  Simple  Machine . . . . 449 

The  Catenary 459 

Friction  between  Cords  and  Cylindrical  Solids 461 

Inclined  Plane. . . . 463 

The  Lever 466 

Wheel  and  Axle 469 

Fixed  Pulley 471 

Movable  Pulley 474 

The  Wedge 480 

The  Screw 484 


viii  CONTENTS. 

PAGE 

Pumps 489 

The  Siphon ...» 489 

The  Air-Pump . 501 

TABLES. 

Table    I. — The  Tenacities  of  Different  Substances,  and  the  Resistances 

which  they  oppose  to  Direct  Compression 508 

"  II. — Of  the  Densities  and  Volumes  of  Water  at  Different  Degrees 
of  Heat  (according  to  Stampfer),  for  every  2\  Degrees  of 
Fahrenheit's  Scale 510 

**     III. — Of  the  Specific  Gravities  of   some  of    the   most  Important 

Bodies »» «.».... 511 

*  IV. — Table  for  finding  Altitudes  with  the  Barometer...... 514 

"      V. — Coefficient  Values,  for  the  Discharge  of  Fluids  through  thin 

Plates,  the  Orifices  being  Remote  from  the  Lateral  Faces 

of  the  Vessel.. 510 

*  VI. — Experiments  on  Friction,  without  Unguents.  By  M.  Morin.  517 
kx  VII. — Experiments  on  Friction  of  Unctuous  Surfaces,  By  M.  Morin.  520 
"VIII. — Experiments  on  Friction  with  Unguents  interposed.    By  M.. 

Morin.. 521 

"     IX. — Friction  of  Trunnions  in  their  Boxes.   ...................  528 

*'      X — Of  Weights   necessary  to   Bend   different    Ropes  around  a 

Wheel  one  Foot  in  Diameter. ..........................  524 


The  Greek  Alphabet  is  here  inserted  to  aid  those  who  are  not  already  famil 
far  with  it,  in  reading  the  parts  of  the  text  in  which  its  letters  occurw 


Letters. 

Names. 

A  a 

Alpha 

b  jse 

Beta 

r  yf 

Gamma 

A  6 

Delta 

E  s 

Epsilon 

z  a 

Zeta 

Uri 

Eta 

0    &d 

Theta 

I     1 

Iota 

K  x 

Kappa 

A  X 

Lambda 

Mp 

111 

Letters. 

Names 

N  v 

Nu 

«  I 

Xi 

O  o 

O  micron 

n  #*• 

Pi 

p  p* 

Rho 

£  ds 

Sigma 

T  rl 

Tau 

T  u 

Upsilon 

$  <p 

Phi 

XX 

Chi 

*  + 

Psi 

ft   W 

Omega 

ELEMENTS 


OF 


ANALYTICAL  MECHANICS. 


INTRODUCTION,. 

PHYSICAL  SCIENCE. 

§  1. — Of  the  existence  of  bodies,  we  are  rendered  conscious  by 
the  impressions  they  make  upon  the  mind  through  the  senses. 

The  condition  of  every  body  is  subject  to  a  variety  of  changes. 
These  changes  are  brought  about  by  agents  external  to  the  bodies 
themselves;  and  to  investigate  nature  with  reference  to  these 
changes  and  their  causes,  is  the  object  of  Physical  Science. 

§  2. — Extension  is  that  by  which  every  body  occupies  a  limited 
portion  of  space.    From  it  the  body  derives  its  figure  and  volume. 

Impenetrability  is  that  which  prevents  two  bodies  from  occupy- 
ing the  same  space  at  the  same  time.  It  determines  a  body's 
identity. 

A  body,  then,  is  any  thing  which  has  extension  and  impene- 
trability. 

§  3. — Elasticity  is  that  property  by  which  a  body  resumes  of 
itself  its  figure  and  dimensions,  when  these  have  been  changed  or 
altered  by  any  extraneous  cause.  Different  bodies  possess  this 
property  in  very  different  degrees,  and  retain  it  with  very  unequal 
tenacity.  Glass,  tempered  steel,  ivory  and  whalebone,  are  among 
the  more  elastic  solids.     All  fluids  are  highly  elastic. 


14:         ELEMENTS    OF    ANALYTICAL    MECHANICS. 


REST,   MOTION,  FORCE. 

§  4. — The  state  of  a  body  by  which  it  continues  in  the  same 
place,  is  called  rest;  that  by  which  it  passes  from  one  place  to 
another,  is  called  motion ;  and  whatever  changes  the  state  of  a 
body  or  the  elements  of  a  body,  with  respect  to  rest  or  motion,  is 
called  force.  The  existence  of  force  is  inferred  from  the  changes, 
with  respect  to  rest  or  motion,  which  all  bodies  and  their  internal 
elements  are  found  to  be  continually  undergoing.  Its  nature,  or 
in  what  it  consists,  is  unknown. 


I     <■     <        C     C 
•      o      *     »   .    «      « 

t  <     c  *,   <   ( 


§  5. — We  shall  find,  though  not  always  upon  superficial  inspec- 
/!  &6q,  £h£f  the  approaching  and  receding  of  bodies  or  of  their  com- 
ponent parts,  when  this  takes  place  apparently  of  their  own 
accord,  are  but  the  results  produced  by  the  various  forces  that 
•come  under  our  notice.  In  other  words,  that  the  universally 
operating  forces  are  those  of  attraction  and  of  repulsion. 

§  6. — Experience  proves  that  these  universal  forces  are  at  work 
in  two  essentially  different  modes.  They  are  operating  either  in 
the  interior  of  a  body,  amidst  the  elements  which  compose  it,  or 
they  extend  their  influence  through  a  wide  range,  and  act  upon 
bodies  in  the  aggregate. 


CONSTITUTION  OF  BODIES. 

§  7. — It  is  assumed  that  all  matter  consists  of  indivisible  atoms. 
And  these  atoms  are  endowed  with  attractive  and  repulsive  forces, 
varying  both  in  intensity  and  direction  by  a  change  of  distance, 
so  that  at  one  distance  two  atoms  attract  each  other,  and  at 
another  distance  they  repel. 

This  law  of  variation  is  the  same  in  all  atoms.  It  is,  there- 
fore, mutual;  for  the  distance  of  atom  a  from  atom  b,  being  the 
.same  as  that  of  b  from  a,  if  a  attract  b.  b  must  attract  a  with 
precisely  the  same  force. 


INTRODUCTION.  15 

Two  or  more  atoms  may  be  so  situated,  in  respect  to  position 
and  distance,  as  to  constitute  a  molecule.  Two  or  more  molecules 
may  constitute  a  particle.    The  particles  constitute  a  body. 

§  8. — The  force  which  is  exerted  between  twro  molecules  when 
cheir  distance  is  diminished  without  end,  and  is  just  vanishing, 
is  an  insuperable  repulsion,  so  that  no  force  whatever  can  press 
two  molecules  into  mathematical  contact. 

§  9. — The  molecular  forces,  here  considered,  are  the  effective 
causes  which  determine  a  body  to  be  a  solid,  liquid,  or  gas. 

§  10. — The  molecular  forces  may  so  act  upon  the  elements  of 
dissimilar  bodies  as  to  cause  a  new  combination  or  union  of  their 
atoms,  This  may  also  produce  a  separation  between  the  combined 
atoms  or  molecules,  in  such  manner  as  to  entirely  change  the 
individual  properties  of  the  bodies.  Such  efforts  of  the  molecular 
forces  are  called  chemical  action;  and  the  disposition  to  exert 
these  efforts,  chemical  affinity. 

§  11. — At  all  considerable  or  sensible  distances,  these  mutual 
forces  are  attractive  and  sensibly  proportional  to  the  square,  of 
the  distance  inversely.     This  attraction  is  called  gravitation. 

§  12. — Our  limits  will  not  permit  us  to  dwell  upon  these  points, 
but  we  cannot  dismiss  the  subject  without  suggesting  one  of  its 
most  interesting  consequences. 

According  to  the  highest  authority,  the  sun  and  other  heavenly 
bodies  have  been  formed  by  the  gradual  subsidence  of  •  a  vast 
nebula,   toward   its   centre.     Its   molecules,  forced   by    their  gravi- 

*  t 

feting  action  within  their  neutral  limits,  are  in  a  state  of  tension, 
which  is  the  more  intense  as  the  accumulation  is  greater;  and 
the  molecular  agitations  in  the  sun,  maintained  by  the  successive 
depositions  at  its  surface,  make  this  body,  in  consequence  of  its 
vast  size,  a  perpetual  fountain  of  that  incessant  stream  of  ethereal 
waves  which  constitute  the  essence  of  light  and  heat.    The  origin 


16        ELEMENTS    OF    ANALYTICAL    MECHANICS. 

of  the  internal  heat  of  the  earth  has  the  same  explanation.  All 
bodies  would  appear  self-luminous  were  the  range  of  our  sense  of 
sight  increased  beyond  its  present  limit  in  the  same  proportion 
that  the  sun  exceeds  them  in  size.  The  sun  far  transcends  all  the 
other  bodies  of  our  system  in  regard  to  heat  and  light,  and  is  in 
a  state  of  incandescence,  because  of  the  mode  of  its  formation  and 
of  its  vastly  greater  dimensions. 

§  13. — The  term  universal  gravitation  is  employed  when  it  is 
intended  to  express  the  action  of  the  heavenly  bodies  on  each 
other;  and  that  of  terrestrial  gravitation  or  simply  gravity,  where 
we  wish  to  express  the  action  of  the  earth  upon  the  bodies  form- 
ing with  itself  one  whole.  The  force  is  always  of  the  same  kind, 
however,  and  varies  in  intensity  only  by  reason  of  a  difference  in 
the  number  of  atoms  and  their  distances.  Its  effect  is  always  to 
generate  motion  when  the  bodies  are  free  to  move. 

Gravity,  then,  is  a  property  common  to  all  terrestrial  bodies, 
since  they  constantly  exhibit  a  tendency  to  approach  the  earth 
and  its  centre.  In  consequence  of  this  tendency,  all  bodies,  unless 
Supported,  fall  to  the  surface  of  the  earth,  and  if  prevented  by 
any  other  bodies  from  doing  so,  they  exert  a  pressure  on  these 
latter. 

This  is  one  of  the  most  important  properties  of  terrestrial 
bodies,  and  the  cause  of  many  phenomena,  of  which  a  fuller 
explanation  will  be  given  hereafter. 


INERTIA. 

§  14. — Experience  shows  that  all  bodies  possess  a  common 
property,  by  which  they  resist  of  themselves  every  change  of 
their  own  state  in  regard  to  rest  or  motion,  and  with  a  force 
equal  to  that  which  produces  the  change.  This  property  is  called 
Inertia,  and  as  a  force  is  both  passive  and  conservative. 

§  15. — Now,  if  to  the  centres  or  loci  of  the  elements  of  what 
is  termed  matter  we  attribute  the  property  called  inertia,  we  have 


INTRODUCTION.  17 

all  the  conditions  requisite  to  explain,  or  arrange  in  the  order  of 
antecedent  and  consequent,  the  various  operations  of  the  physical 
world. 

Whenever  any  force  acts  upon  a  free  body,  the  inertia  of  the 
latter  reacts,  and  this  action  and  reaction  are  equal  and  contrary. 

§  16. — Mechanics  is  founded  upon  a  single  fact,  viz. :  that  all 
action  is  ever  accompanied  by  an  equal,  contrary,  and  simultaneous 
reaction. 

MASS. 

§  17. — The  mass  of  a  body  is  the  quantity  of  matter  it  con- 
tains; and  this  being  proportional  to  its  weight,  the  mass  of  a 
body  may  be  measured  by  the  quotient  arising  from  dividing  its 
weight  by  the  weight  of  some  other  body  assumed  as  the  unit  of 
mass.  A  cubic  foot  of  distilled  water,  at  its  maximum  density, 
has  been  assumed  as  the  unit  of  mass.  And,  therefore,  a  body 
whose  mass  is  expressed  by  any  number,  say..  20,  will  contain 
twenty  times  the  matter  contained  in  a  Cubic  foot  of  distilled 
water  at  its  greatest  density. 

!  §  18. — Density  is  a  term  employed  to  denote  the  degree  of 
proximity  of  the  atoms  of  a  body.  Its  measure  is  the  ratio 
arising  from  dividing  the  weight  of  the  body  by  the  weight  of  an 
ejuai  volume  of  some  standard  substance  whose  density  is  assumed 
as  unity. 

If  the  mass  of  a  body  be  denoted  by  M,  its  weight  by  W,  and 
that  of  a  unit  of  mass  by  g,  then  will 

W—M.g (1) 

If  V  denote  the  body's  volume,  and  D  its  density ;   then  will 

M=  V.D (2) 

and  by  substitution  above, 

W=V.D.g 


18  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Equation  (1)  is  applicable  only  in  problems  where  W  and  g 
are  taken  in  the  sense  above  given.  We  shall  find,  however,  that 
there  is  another  measure  for  the  force  of  gravity  in  terms  of  what 
is  called  quantity  of  motion.  It  is,  therefore,  to  be  borne  in  mind, 
that  when  dealing  with  questions  which  involve  the  action  of  the 
force  of  gravity,  we  must  distinguish  between  those  cases  where 
the  effects  are  simple  pressures  and  those  where  the  effects  are 
motions.  In  the  first  case  the  symbol  g  represents  62.5  pounds 
and  in  the  second  a  velocity  of  32.18  ±  feet  per  second^the 
expression  for  the  mass  always  being  an  abstract  number  or  ratio, 
and  the  same  in  both. 

MECHANICS. 

§  19. — All  phenomena  of  the  physical  world  arise  directly  from 
the  action  of  forces  upon  the  various  forms  of  bodies.  That 
branch  of  science  which  treats  of  this  action  is  called  Mechanics. 
A  careful  study  of  mechanics  is,  therefore,  an  indispensable  prep- 
aration for  that  of  any  branch  of  physical  science.  Mechanics  is 
the  subject  of  the  present  volume.  It  will  be  treated  under  three 
heads,  suggested  by  peculiarities  of  physical  condition,  viz.: 
Mechanics  of  Solids,  Mechanics  of  Fluids,  and  Mechanics  of 
Molecules ;  the  first  treating  of  the  action  of  forces  upon  solid 
bodies;  the  second,  upon  fluid  bodies;  and  the  third,  upon  the 
molecules  or  elements  of  both  solids  and  fluids. 


PART    I. 


MECHANICS  OF  SOLIDS. 


FORCE    AND    MOTION 

§  20. — Of  the  nature  of  force  we  know  only  that  it  is  of  spiritual 
origin,  created  with  matter  by  that  Divine  Ruler  whose  almighty  will 
is  the  source  of  the  universe  and  of  the  laws  which  it  obeys. 

Those  laws  our  finite  powers  are  sufficient  to  enable  us  to  investi- 
gate, though  imperfectly.  By  obeying,  we  can  command  and  use  for 
our  benefit,  or  pleasure.  By  violating,  we  are  generally  made  to 
suffer. 

§  21. — The  study  of  physical  science  is  only  the  study  of  the 
effects  produced  by  forces,  and  of  their  laws  or  modes  of  action, 
which  is  all  that  we  can  know  of  them. 

When  we  pull  or  push  a  body,  we  become  conscious  both  of  the 
force,  muscular  strain,  effort,  or  pressure  exerted  by  ourselves,  in 
obedience  to  our  will,  and  of  the  resistance  which  the  body  opposes 
to  our  action,  while  it  either  remains  at  rest  or  undergoes  any  change 
of  motion.  That  resistance  is  itself  force,  and  it  is  the  equal  and 
contrary  reaction  which  ever  takes  place,  or  the  inertia  of  the  matter 
composing  the  body.  From  such  personal  experience  wre  derive  our 
conception  of  what  is  called  force?  the  unknown  cause  of  effects 
observed. 


20         ELEMENTS    OF    ANALYTICAL    MECHANICS 

§  22. — But  force  exists  in  other  forms  than  those  of  muscular 
action  and  inertia;  and  its  identity  under  its  various  modes  of  man- 
ifestation has  been  matter  of  much  research  and  discussion.  When 
atoms  combine  by  what  is  called  chemical  affinity  to  constitute  a 
molecule,  as  when  hydrogen  and  oxygen  unite  to  form  water,  this  is 
but  the  result  of  their  mutual  attraction.  And  when  heat,  or  elec- 
tricity, act  to  decompose  water  into  its  elements,  hydrogen  and 
oxygen,  this  analysis,  or  dissociation,  is  only  the  result  of  mutual 
repulsions.  So  also  whenever  a  body  is  expanded  in  volume  by  heat, 
that  heat  is  only  the  mutual  repulsive  force  pushing  the  molecules 
away  from  each  other. 

Of  late  years,  it  has  been  generally  admitted  to  have  been 
proved  by  abundant  evidence  that  all  the  physical  forces  are  essen- 
tially one,  or  identical.  And,  consequently,  that  any  given  effect  of 
force  is  capable  of  being  changed  into  an  equivalent  effect  of  a 
totally  different  kind.  Thus,  mechanical  force,  gravitation,  heat,  light, 
electricity,  magnetism,  and  chemical  affinity  are  all  readily  trans- 
mutable,  or  convertible,  into  each  other,  and  constitute  only  different 
modes  of  manifestation  of  that  one  force,  to  which  the  name  of 
energy  has  been  applied  to  denote  its  unity  and  their  identity. 

§  23. — Observation  appears  also  to  prove  that  though  energy  is 
capable  of  change  into  various  equivalent  forms,  it  is  indestructible. 
It  may  be  given  by  one  body  to  another,  as  when  a  moving  body 
strikes  and  puts  another  in  motion,  or  a  hot  body  heats  a  colder 
one;,  but  this  transference  is  not  attended  with  any  destruction. 
Thus  power,  force,  or  energy,  like  matter,  is  held  to  be  incapable  of 
loss,  or  annihilation.  Created  by  God,  no  finite  power  is  sufficient 
to  destroy  what  Omnipotence  brought  into  and  keeps  in  being.  This 
induction  of  modern  science,  founded  upon  very  extended  and  thor- 
ough investigation,  is  now  called  the  law  of  the  conservation  of  energy, 
because  it  asserts  the  indestructibility  of  energy. 

§  24. — From  the  above,  it  is  evident  that  there  are  only  two  ways 
in  which  atoms,  molecules  and  masses  act  upon  each  other,  by 
attraction  or  repulsion ;    or,  in    other  words,  they  either  pull    or  push. 


MECHANICS    OF    SOLIDS 


21 


each  other  with  equal  force ;  and  this  is  what  is  meant  by  the  law 
of  equal  action  and  reaction.  It  is  also  evident  that  there  is  always 
a  duality  in  the  action  of  force,  one  atom,  molecule,  or  mass,  never 
being  pulled  or  pushed  except  by  another  atom,  molecule,  or  mass. 
Thus  the  exertion  of  force  is  found  to  be  inseparable  from  the  mat- 
ter from  which  it  emanates,  and  incapable  of  manifestation  except 
by  the  mutual  action  and  reaction  of  two  portions  of  matter. 

§  25. — Formerly,  in  Mechanics,  the  manner  in  which  forces  act 
upon  solids  was  usually  considered  to  be  that  of  simultaneous  appli- 
cation to  each  and  every  portion  of  a  perfectly  rigid  mass,  whose 
molecules  were  not  regarded  as  separated  in  space,  but  as  in  contact 
and  forming  continuous  parts  of  one  geometric  volume  or  whole. 
This  hypothesis  of  continuous  extension  and  complete  rigidity,  to 
which  the  o-eometric  methods  of  differentiation  and  integration  of 
volumes  were  of  easy  application,  while  it  answered  well  for  the  uses 
of  Astronomy,  was  not  adapted  to  many  parts  of  terrestrial  Mechanics. 
The  development  of  the  mechanics  of  molecules,  chiefly  by  Fresnel, 
Navier,  and  Cauchy,  and  the  subsequent  extension  of  that  branch  of 
science  by  others,  has  led  to  very  improved  modes  of  regarding  and 
investigating  the  action  of  forces  upon  masses. 


§  2G.~ The    manner  in  which    forces   geneially    act   has    been   so 
admirably  set   forth    by  Poncelet,  that    we 
shall  here    give   a   translation    of   much  of 
the  language  he  has  employed : 

If  a  free  body  be  drawn  by  a  thread, 
the  thread  will  stretch  and  even  break  if 
the  action  be  too  violent,  and  this  will  the 
more  probably  happen  in  proportion  as  the 
body  is  more  massive.  If  a  body  be 
suspended  by  means  of  a  vertical  chain, 
and  a  weighing-spring  be  interposed  in 
the  line  of  traction,  the  graduated  scale 
of  the  spring  will  indicate  the  weight  of  - 
the  body  when    the    latter  is   at  rest;    but   if   the    upper  end    of   the 


22         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

chain  be  suddenly  elevated,  the  spring  will  immediately  bend  more 
in  consequence  of  the  resistance  opposed  by  the  inertia  of  the  body 
while  acquiring  motion.  When  the  motion  acquired  becomes  uniform, 
the  spring  will  resume  and  preserve  the  degree  of  flexure  which  it 
had  at  rest.  If  now  the  motion  be  checked  by  relaxing  the  effort 
applied  to  the  upper  end  of  the  chain,  the  spring  will  unbend  and 
indicate  a  pressure  less  than  the  weight  of  the  body,  in  consequence 
of  the  inertia  acting  in  opposition  to  the  retardation.  The  oscilla- 
tions of  the  spring  may  therefore  serve  to  indicate  the  variations  in 
the  motions  of  a  body  and  the  energy  of  its  force  of  inertia,  which 
acts  against  or  with  a  force,  according  as  the  velocity  is  increased  or 
diminished. 

§  27. — Forces  produce  various  effects,  according  to  circumstances. 
They  sometimes  leave  a  body  at  rest,  by  balancing  one  another, 
through  its  intervention ;  sometimes  they  change  its  form  or  break 
it;  sometimes  they  impress  upon  it  motion,  they  accelerate  or  retard 
that  which  it  has,  or  change  its  direction  ;  sometimes  these  effects  arc 
produced  gradually,  sometimes  abruptly,  but  however  produced,  they 
require  some  definite  time,  and  are  effected  by  continuous  degrees.  If 
a  body  is  sometimes  seen  to  change  suddenly  its  state,  either  in 
respect  to  the  direction  or  the  rate  of  its  motion,  it  is  because  the 
force  is  so  great  as  to  produce  its  effect  in  a  time  so  short  as  to 
make  its  duration  imperceptible  to  our  senses,  yet  some  definite  por- 
tion of  time  is  necessary  for  the  change.*  A  ball  fired  from  a  gun 
will  break  through  a  pane  of  glass,  a  piece  of  board,  or  a  sheet  of 
paper,  when  freely  suspended,  with  a  rapidity  so  great  as  to  call  into 
action  a  force  of  inertia  in  the  parts  which  remain,  greater  than  the 
molecular  forces  which  connect  the  latter  with  those  torn  away. 

In  such  cases  the  effects  are  obvious,  while  the  times  in  which 
they  are  accomplished  are  so  short  as  to  elude  the  senses;  and  yet 
these  times  have  had  some  definite  duration,  since  the  changes  cor- 
responding to  these  effects  have  passed  in  succession  through  their 
different  degrees  from  the  beginning  to  the  ending. 

§  28. — The    statical    measure    of   forces   may    be    obtained    by   an 


MECHANICS    OF    SOLIDS. 


23 


instrument  called  the  Dynamometer,  which  in  principle  does  not  differ 
from    the    spring  balance.     The  dynamical 
measure  will  be  explained  further  on. 


8  29. — When  a  force  acts  against  a 
point  in  the  surface  of  a  body,  it  exerts 
a  pressure  which  crowds  together  the  neigh- 
boring particles;  the  body  yields,  is  com- 
pressed and  its  surface  indented ;  the 
crowded  particles  make  an  effort,  by  their 
molecular  forces,  to  regain  their  primitive 
places,  and  thus  transmit  this  crowding 
action    even   to   the    remotest    particles   of 

the  body.  If  these  latter  particles  are  fixed  or  prevented  by  obstacles 
from  moving,  the  result  will  be  a  compression  and  change  of  figure 
throughout  the  body.  If,  on  the  contrary,  these  extreme  particles 
are  free,  they  will  advance,  and  motion  will  be  communicated  by 
degrees  to  all  parts  of  the  body.  This  internal  motion,  the  result 
of  a  series  of  compressions,  proves  that  a  certain  time  is  necessary 
for  a  force  to  produce  its  entire  effect,  and  the  error  of  supposing 
that  a  finite  velocity  may  be  generated  instantaneously.  The  same 
kind  of  action  will  take  place  when  the  force  is  employed  to  destroy 
the  motion  which  a  body  has  already  acquired ;  it  will  first  destroy 
the  motion  of  the  molecules  at  and  nearest  the  point  of  action,  and 
then,  by  degrees,  that  of  those  which  are  more  remote  in  the  order 
of  distance. 

The  molecular  springs  cannot  be  compressed  without  reacting  in  a 


oaM» 


contrary  direction,  and  with  an  equal  effort.  The  agent  which  presses 
a  body  will  experience  an  equal  pressure;  reaction  is  equal  and  con- 
trary to  action.     In  pressing   the    finger  against  a   body,  in  pulling  it 


24        ELEMENTS    OF    ANALYTICAL    MECHANICS. 

with  a  thread,  or  pushing  it  with  a  bar,  we  are  pressed,  drawn,  or 
pushed  in  a  contrary  direction,  and  with  an  equal  effort.  Two  weigh- 
ing springs  attached  to  the  extremities  of  a  chain  or  bar  will  indicate 
the  same  degree  of  tension  and  in  contrary  directions  when  made  to 
act  upon  each  other  through  its  intervention. 

In  everv  case,  therefore,  the  action  of  a  force  is  transmitted  through 
a  body  to  the  ultimate  point  of  resistance  by  a  series  of  equal  and 
contrary  actions  and  reactions  which  neutralize  each  other,  and  which 
the  molecular  springs  of  all  bodies  exert  at  every  point  of  the  right 
line  along  which  the  force  acts.  It  is  in  virtue  of  this  property  of 
bodies  that  the  action  of  a  force  may  be  assumed  to  be  excited  at 
any  point  in  its  line  of  direction  within  the  boundary  of  the  body. 

Bodies  being  more  or  less  extensible  and  compressible,  when 
interposed  between  the  force  and  resistance  will  be  stretched  or  com- 
pressed to  a  certain  degree,  depending  upon  the  energy  with  which 
these  forces  act;  but  as  long  as  the  force  and  resistance  remain  the 
same,  the  body  having  attained  its  new  dimensions  will  cease  to 
change.  On  this  account  we  may,  in  the  investigations  which  follow, 
assume  that  the  bodies  employed  to  transmit  the  action  of  forces  from 
one  point  to  another  are  inextensible  and  rigid. 

i. 
§  30. — The  intensity  of   a   force  is  its   greater  or  less  capacity  to 

produce    pressure    or    strain.     This    intensity    may    be    expressed    in 

pounds,  or  in  quantity  of   motion.     Its  value    in    pounds  is  called   its 

statical  measure;    in  quantity  of  motion,  its  dynamical  measure. 

§  31. — The  point  of  application  of  a  force  is  the  material  point  to 
which  the  force  may  be  regarded  as  directly  applied. 

§  32. — The  line  of  direction  of  a  force  is  the  right  line  which  the 
point  of  application  would  describe,  if  it  were  perfectly  free  and 
moved  from  rest. 

§  33. — -The  effect  of  a  force  depends  upon  its  intensity,  point  of 
application,  and  line  of  direction,  and  when  these  are  given  the  force 
is  known. 


MECHANICS    OF    SOLIDS.  25 

§  34. — Two  forces  are  equal  when,  substituted  one  for  the  other, 
in  the  same  circumstances,  they  produce  the  same  effect,  or  when, 
directly  opposed,  they  neutralize  each  other. 


MOTION    AND    REST. 

§  35. — A  body  would  be  absolutely  at  rest  if  it  should  remain  in 
the  same  place  and  position  in  space.  And  it  is  in  motion  if  it 
changes  its  position.  All  bodies  in  nature  appear  to  be  in  motion. 
Rest  is,  therefore,  only  a  relative  term,  and  so  also  are  the  usual 
motions  which  we  observe.  They  seem  to  us  absolute  only  because 
we  refer  them  to  points  considered  as  fixed.  This  is  due  to  their 
being  seen  by  us  at  the  surface  of  the  earth,  where  all  objects  par- 
take of  the  same  motions  around  the  earth's  axis  and  about  the  sun. 
A  body  which  continues  in  the  same  place  in  a  boat  is  said  to  be 
at  rest  in  relation  to  the  boat,  although  the  boat  itself  may  be  in 
motion  in  relation  to  the  banks  of  a  river  on  whose  surface  it  is 
floating;.  I 

§  36. — Motion  is  essentially  continuous ;  that  is,  a  body  cannot 
pass  from  one  position  to  another  without  passing  through  a  series 
of  intermediate  positions;  a  point  in  motion,  therefore,  describes  a 
continuous  line. 

When  we  speak  of  the  path  described  by  a  body,  we  are  to 
understand  that  of  a  certain  point  connected  with  the  body.  Thus, 
the  path  of  a  ball  is  that  of  its  centre. 

§  37. — The  motion  of  a  body  is  said  to  be  curvilinear  or  rectilinear, 
according  as  the  path  described  is  a  curve  or  right  line.  Motion  is 
either  uniform  or  varied.  A  body  is  said  to  have  uniform  motion 
when  it  passes  over  equal  spaces  in  equal  successive  portions  of  time : 
and  it  is  said  to  have  varied  motion  when  it  passes  over  unequal 
spaces  in  equal  successive  portions  of  time.  The  motion  is  said  to 
be  accelerated  when  the  successive  increments  of  space  in  equal  times 
become  greater  and  greater.  It  is  retarded  when  these  increments 
become  smaller  and  smaller. 


26         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

§  38. —  Velocity  is  the  rate  of  a  body's  motion.  Velocity  is 
measured  by  the  length  of  path  described  uniformly  in  a  unit  of 
time. 

g  39. — The  spaces  described  in  equal  successive  portions  of   time 

being  equal   in    uniform    motion,  it   is   plain  that  the   length  of   path 

described   in    any  time  will    be    equal  to  that  described  in  a  unit  of 

time  repeated   as  many  times  as   there    are  units  in  the  time.     Let  v 

denote  the  velocity,  t   the  time,  and   s  the  length  of   path  described, 

then  will 

$  =  v.  t (3) 

If  the  position  of  the  body  be  referred  to  any  assumed  origin 
whose  distance  from  the  point  where  the  motion  begins,  estimated  in 
the  direction  of  the  path  described,  be  denoted  by   S,  then  will 

s  =  S  -f  v  A .     .     (4) 

Equation  (3)  shows  that  in  uniform  motion,  the  space  described 
is  always  equal  to  the  product  of  the  time  into  the  velocity ;  that 
the  spaces  described  by  different  bodies  moving  with  different  velocities 
during  the  same  time,  are  to  each  other  as  the  velocities ;  and  that  lohen 
the  velocities  are  the  same,  the  spaces  are  to  each  other  as  the  times. 

§  40. — Differentiating  Equation  (3)  or  (4),  we  find 

ds  .  . 

3?-" <5) 

that  is  to  say,  the   velocity  is  equal   to   the  first  differential   coefficient 
of  the  space  regarded  as  a  function  of  the  time. 

Dividing  both  members  of  Equation  (3)  by  t,  we  have 

7  =  ' <6> 

which  shows  that,  in  uniform  motion,  the  velocity  is  equal  to  the  whole 
space  divided  by  the  time  in  which  it  is  described. 

WORK. 

§  41. — To  work  is  to  overcome  a  resistance  continually  recurring 
along  some  path.     Thus,  to    raise    a  body  through    a  vertical    height, 


MECHANICS    OF    SOLIDS.  27 

its  weight  must  be  overcome  at  every  point  of  the  vertical  path.  If 
a  body  fall  through  a  vertical  height,  its  weight  develops  its  inertia 
at  every  point  of  the  descent.  To  take  a  shaving  from  a  board  with 
a  plane,  the  cohesion  of  the  wood  must  be  overcome  at  every  point 
along  the  entire  length  of  the  path  described  by  the  edge  of  the 
chisel. 

§  42. — The  force  may  be  constant,  or  it  may  be  variable.  In  the 
first  case,  the  quantity  of  work  performed  is  the  constant  force  taken 
as  many  times  as  there  are  points  at  which  it  has  acted,  and  is 
measured  by  the  product  of  the  force  into  the  path  described  by  its 
point  of  application,  estimated  in  the  direction  of  the  force.  When 
the  force  is  variable,  the  quantity  of  work  is  obtained  by  estimating 
the  elementary  quantities  of  work  and  taking  their  sum.  By  the 
elementary  quantity  of  work  is  meant  the  intensity  of  the  variable 
force  taken  as  many  times  as  there  are  points  in  the  indefinitelv 
small  path  over  which  the  force  may  be  regarded  as  constant;  and 
is  measured  by  the  intensity  of  the  force  into  the  differential  of  the 
path,  estimated  in  the  direction  of  the  force. 

§  43. — In  general,  let  P  denote  the  intensity  of  any  variable  force 

and  s  the  path  described  by  its  point  of  application,  estimated  in  the 

direction  of  the  force;   then  will  the  quantity  of  work,  denoted  by  Q, 

be  given  by 

Q  =  fP.ds (V) 

which,  integrated  between  certain  limits,  will  give  the  value  of   Q. 

§  44. — The  simplest  kind  of  work  is  that  performed  in  raising  a 
weight  through  a  vertical  height.  It  is  taken  as  a  standard  of  com- 
parison, and  suggests  at  once  an  idea  of  the  quantity  of  work 
expended  in  any  particular  case. 

Let  the  weight  be  denoted  by  PP,  and  the  vertical    height  by  H\ 

then  will 

Q=W.H (8) 

If  W  become  one  pound,  and  H  one  foot,  then  will 


28 


ELEMENTS    OF    ANALYTICAL    MECHANICS 


and  the  unit  of  work  is,  therefore,  the  unit  of  force,  one  pound, 
exerted  over  the  unit  of  distance,  one  foot;  and  is  represented  by  a 
square  of  which  the  adjacent  sides  are  respectively  one  foot  and  one 
pound,  taken  from  the  same  scale  of  equal  parts. 


§  45. — To  illustrate  the  use  of  Equation  (V),  let 
it  be  required  to  compute  the  quantity  of  work 
necessary  to  compress  the  spiral  spring  of  the 
common  spring  balance  to  any  given  degree,  say 
from  the  length  AB  to  DB.  Let  the  resistance 
vary  directly  as  the  degree  of  compression,  and 
denote  the  distance  AD'  by  x\   then  will 

P  =  C.x; 

in   which  C  denotes  the  resistance   of  the  spring 

when  the  balance  is  compressed  through  the  dis- 

i 
tance  unity. 

This  value  of  P  in  equation  (7)  gives 

Q=fP.dx  =  fC.  xdx  =  tf.~  +  C, 

25 

which,  integrated  between  the  limits  x  =  0  and  x  =  AD  =  a,  gives 

Let  C=  10  pounds,  a  =  3  feet;    then  will 

Q  =  45  units  of  work, 

and  the  quantity  of  work  will  be  equal  to  that  required  to  raise 
45  pounds  through  a  vertical  height  of  one  foot,  or  one  pound 
through  a  height  of  45  feet,  or  9  pounds  through  5  feet,  or  5  pounds 
through  9  feet,  <fcc,  all  of  which  amounts  to  the  same  thing.  The 
work  of  a  resistance  is  numerically  equal  to  that  of  the  force  which 
develops  it,  but  taken  with  contrary  Burn. 

§  46.— A  mean  resistance  is  that  which,  multiplied  into  the  entire 
path  described  in  the  direction  of  the  resistance,  will  o-ive  the  entire 


MECHANICS    OF    SOLIDS 


29 


quantity  of  work.     Denote  its  intensity  by  i?,  and  the  entire  path  by 
*,  and  from  the  definition  we  have 


whence, 


B.s  =  /P.ds; 


(9). 


That  is,  the  mean  resistance  is  equal  to  the  entire  work,  divided 
by  the  entire  path. 

In  the  above  example  the  path  being  3  feet,  the  mean  resistance 
would  be  15  pounds. 


§  47. — Equation  (7)  shows  that  the  quantity  of  work  can  be  rep- 
resented by  the  area  included  between  the  path  s,  in  the  direction 
of  the  resistance,  the  curve  whose  ordinates  are  the  different  values 
of  Py  and  the  ordinates  which  denote  the  extreme  resistances.  When- 
ever, therefore,  the  curve  which  connects  the  resistance  with  the  path 
is  known,  the  process  for  finding  the  quantity  of  work  is  one  of  sim- 
ple integration. 

Sometimes  this    law    cannot    be   found,  and    the    intensity    of   the 
resistance  is  given   only   at  certain   points  of  the  path.     In  this  case 
we    proceed    as   follows,    viz. :     At   the    several    points    of    the    path 
where  the  resistance  is  known, 
erect   ordinates    equal   to  the 
corresponding  resistances,  and 
connect   their   extremities   by 
a   curved    line;     then    divide 
the    path    described   into  any 
even   number   of   equal  parts, 
and  erect  the  ordinates  at  the 
points  of  division,  and  at  the 
extremities ;  number  the  ordi- 
nates in  the  order  of  the  nat- 
ural numbers ;    add  together  the  extreme  ordinates,  increase  this  sum  by 
four    times    that   of  the   even   ordinates   and    twice  that  of  the  uneven 
ordinates,  and   multiply  by  one  third  of  the  distance  between  any  two 
consecutive  ordinates. 


1 


% 


e 
o 


e 


7 


30    ELEMENTS  OF  ANALYTICAL  MECHANICS. 

The  demonstration  of  this  rule  is  analogous  to  that  employed  in 
Mensuration,  in  computing  the  approximate  area  bounded  by  the 
curve  7*1  r7,  the  right  line  exeni  and  the  two  extreme  ordinates. 

§  48. — By  the  processes  now  explained,  it  is  easy  to  estimate  the 
quantity  of  work  of  the  weights  of  bodies,  of  the  resistances  due  to 
the  forces  of  affinity  which  hold  their  elements  together,  of  their 
elasticity,  <fcc.  It  remains  to  consider  the  rules  by  which  the  quantity 
of  work  of  inertia  may  be  computed.  Inertia  is  exerted  only  during 
a  change  of  state  in  respect  to  motion  or  rest,  and  this  brings  us  to 
the  subject  of  varied  motion. 

VARIED    MOTION. 

§  49. — Varied  motion  has  been  defined  to  be  that  in  which  unequal 
spaces  are  described  in  equal  successive  portions  of  time.  In  this 
kind  of  motion  the  velocity  is  ever  varying.  It  is  measured  at  any 
given  instant  by  the  length  of  path  it  would  enable  a  body  to 
describe  in  the  first  subsequent  unit  of  time,  were  it  to  remain 
unchanged.  Denote  the  space  described  by  s,  and  the  time  of  its 
description  by  t. 

However  variable  the  motion,  the  space  described  in  the  unit  of 
time  would,  were  the  velocity  constant,  be  ds  repeated  as  many  times 
as  the  unit  of  time  contains  dt.  Hence,  denoting  the  value  of  the 
velocity  at  any  instant  by  v,  we  have 

9  =  ds  x  -n  5 

at 

or, 

ds  ,     . 

•■-■% <"> 

§  50. — Continual  variation  in  a  body's  velocity  can  only  be  pro- 
duced by  the  incessant  action  of  some  force.  The  body's  inertia 
opposes  an  equal  and  contrary  reaction.  This  reaction  is  directly 
proportional  to  the  mass  of  the  body  and  to  the  amount  of  change 
in  its  velocity;  it  is,  therefore,  directly  proportional  to  the  product 
of  the  mass  into  the  increment  or  decrement  of  the  velocity.  The 
product  of   a  mass  into    a  velocity   represents   a    quantity  of  motiov. 


MECHANICS    OF    SOLIDS.  31 

i 

The  intensity  of  a  motive  force,  at  any  instant,  is  assumed  to  be 
measured  by  the  quantity  of  motion  which  this  intensity  can  generate 
in  a  unit  of  time. 

The  mass  remaining  the  same,  the  velocities  generated  in  equal 
successive  portions  of  time,  by  a  constant  force,  must  be  equal  to 
each  other.  However  a  force  may  varv,  were  it  to  remain  constant, 
it  would  generate  in  a  unit  of  time  a  velocity  equal  to  dv  repeated 
as  many  times  as  dt  is  contained  in  this  unit;  that  is,  the  velocity 
generated  would   be  equal   to 

.      1         dv 
dv  .  —  =  —  : 
dt       dt  ' 

and   denoting   the  intensity  of   the   force  by   P  and  the  mass  by  M, 

we  shall  have 

n        ir   dv  ■ 

P  =  M-Tt <12> 

Again,  differentiating  Equation  (11),  regarding  t  as  the  independent 
variable,  we  get, 

dv=di; 
and  this,  in  Equation   (12),  gives 

p  =  M-% <13> 

From  Equation  (11),  we  conclude  that  in  varied  motion,  the  velocity 
at  any  instant  is  equal  to  the  first  differential  coefficient  of  the  space 
regarded  as  a  function  of  the  time. 

From  Equation  (12),  that  the  intensity  of  any  motive  force,  or  of 
the  inertia  it  develops,  at  any  instant,  is  measured  by  the  product  of 
the  mass  into  the  first  differential  coefficient  of  the  velocity  regarded  as 
a  function  of  the  time. 

And  from  Equation  (13),  that  the  intensity  of  the  motive  force, 
or  of  the  inertia,  is  measured  by  the  product  of  the  mass  into  the 
necond  differential  coefficient  of  the  space  regarded  as  a  function  of 
the  time,. 

§  51. — To  illustrate.     Let  there  be  the  relation 

s  =  at*  +  bfi (14) 

required  the  space  described  in  three  seconds,  the  velocity  at  the  end 


32         ELEMENTS    OF    ANALYTICAL    MECHANICS. 


• 


of   the   third    second,   and  the   intensity   of   the    motive    force   at   the 
same  instant. 

Differentiating  Equation  (14)  twice,  dividing  each  result  by  dt,  and 
multiplying  the  last  by  M,  we  find 

—  =  v  =  Sat2  +  2bt (15) 

(X  v 

M.—  —P  —  M[Qat-\-2b} (16) 

4 

Make  a  =  20  feet,  6  =  10  feet,  and  t—3  seconds;  we  have 
from  Equations  (14),  (15),  and  (16), 

f  =  20  .  33  -f  10  .  32  =  630  feet ; 
v  =  3  .  20 .  32  -f  2  .  10 .  3  =  600  feet ; 
P  =  M(G.20.S  +  2. 10)  =  380.  M. 

That  is  to  say,  the  body  will  move  over  the  distance  630  feet  in 
three  seconds,  will  have  a  velocity  of  600  feet  at  the  end  of  the 
third  second,  and  the  force  will  have  at  that  instant  an  intensity 
capable  of  generating  in  the  mass  M  a  velocity  of  380  feet  in  one 
second,  were  it  to  retain  that  intensity  unchanged. 

§  52. — Dividing  Equations  (12)  and  (13)  by  Jf,  they  give 

^  =  ^ (17) 

M-dfi •    •    *    (18) 

The  first  member  is  the  same  in  both,  and  it  is  obviously  that 
portion  of  the  force's  intensity  which  is  impressed  upon  the  unit  of 
mass.  The  second  member  in  each  is  the  velocity  impressed  in  the 
unit  of  time,  and  is  called  the  acceleration  due  to  the  motive  force. 

§  53. — From  Equation  (11)  we  have, 

ds  =  v.dt (19) 

multiplying  this  and  Equation  (12)  together,  there  will  result, 

P.ds  =  M.v.dv (20) 


md  integrating, 


fP.d^^ (21) 


MECHANICS    OF    SOLIDS.  33 

The  first  member  is  the  quantity  of  work  of  the  motive  force, 
which  is  equal  to  that  of  inertia ;  the  product  M.  v2  is  called  the 
vis  viva  or  living  force  of  the  body  whose  mass  is  M.  Whence,  we 
see  that  the  work  of  inertia  is  equal  to  half  the  living  force  ;  and  the 
living  force  of  a  body  is  double  Hie  quantity  of  work  expended  by  iU 
inertia  while  it  is  acquiring  its  velocity. 

§  54, — If  the  force  become  constant  and  equal  to  F,  the  motion 
will  be  uniformly  varied,  and  we  have,  from  Equation  (18), 

F  _cfts 
M~dft 

Multiplying  by  dt  and  integrating,  we  get 

F  ds 

M't  =  dt+C  =  V+C    ~    ~    *    '    <22> 

and  if  the  body  be  moved  from   rest,  the   velocity   will  be  equal  to 
zero  when  t  is  zero ;   whence  C  =  0,  and 

M't  =  v <23)       . 

Multiplying  Equation  (22)  by  dt,  after  omitting  C  from  it,  and 
integrating  again,  we  find 

and  if  the  body  start  from  the  origin  of  spaees,  C  will  be  zero,  and 

F    ft 

M'2=S <24> 

Making  t  equal  to  one  second,  in  Equations  (23)  and  (24),  and 
dividing  the  last  by  the  first,  we  have 


1         s 

¥  ~  ~v" 

or, 

v  =  2s (25) 

That   is   to   say,  the   velocity  generated  in  the  first  unit  of  time  in 
measured    by   double    the   space    described    in    acquiring    this    velocity 
Equations  (23),  (24),  and  (25)  express  the  laws  of  constant  forces. 


34  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

g  55. — The  dynamical  measure  for  the  intensity  of  a  force,  or  the 
pressure  it  is  capable  of  producing,  is  assumed  to  be  the  effect  this 
pressure  can  produce  in  a  unit  of  time,  this  effect  being  a  quantity 
of  motion,  measured  by  the  product  of  the  mass  into  the  velocity 
generated.  This  assumed  measure  must  not  be  confounded  with  the 
quantity  of  work  of  the  force  while  producing  this  effect.  The 
former  is  the  measure  of  a  single  pressure;  the  latter,  this  pressure 
repeated  as  many  times  as  there  are  points  in  the  path  over  which 
this  pressure  is  exerted. 

Thus,  let  the  body  be  moved  from  A  to 
By  under  the  action  of  a  constant  force,  in 
one  second ;  the  velocity  generated  will, 
Equation  (25),  be  2AB.  Make  BC—2AB, 
and  complete  the  square  BCEE.  BE  will 
be  equal  to  v ;  the  intensity  of  the  force 
will  be  M.v;  and  the  quantity  of  work, 
the  product  of  M.v  by  AB,  or  by  its 
equal  -g-u;  thus  making  the  quantity  of 
work  ^Mv2,  or  the  mass  into  one  half  the 
square  BF\  which  agrees  with  the  result  obtained  from  Equation  (21). 

EQUILIBRIUM. 

§  56. — Equilibrium  is  a  term  employed  to  express  the  state  of 
two  or  more  forces  which  balance  one  another  through  the  interven- 
tion of  the  body  subjected  to  their  simultaneous  action.  When 
applied  to  a  body,  it  means  that  the  state  of  the  body  may  either 
:be  rest  or  uniform  motion. 

§  57. — We  must  be  careful  to  distinguish  between  the  extraneous 
forces  which  act  upon  a  body,  and  the  forces  of  inertia  which  they 
may,  or  may  not,  develop. 

If  a  body  subjected  to  the  simultaneous  action  of  several  extraneous 
forces,  be  at  rest,  or  have  uniform  motion,  the  extraneous  forces 
are  in  equilibrio,  and  the  force  of  inertia  is  not  developed.  If  the 
body  have  varied  motion,  the  extraneous  forces  are  not  in  equilibrio, 


MECHANICS    OF    SOLIDS.  35 

but  develop  forces  of  inertia  which,  with  the  extraneous  forces,  are 
in  equilibrio.  Forces,  therefore,  including  the  force  of  inertia,  are 
ever  in  equilibrio ;  and  the  indication  of  the  presence  or  absence 
of  the  force  of  inertia,  in  any  case,  shows  that  the  body  is  or  is 
not  changing  its  condition  in  respect  to  rest  or  motion.  This  is 
but  a  consequence  of  the  universal  law  that  every  action  is  accom- 
panied by  an  equal  and  contrary  reaction. 

GENERAL  LAW  OF  WORK  AND  ENERGY. 

§  58. — The  extraneous  forces,  called  impressed  forces,  being,  there- 
fore, always  in  equilibrio,  either  among  themselves  or  with  the  forces 
of  inertia,  the  sum  of  the  quantities  of  work  performed  in  any  one 
direction,  regarded  as  positive,  must  be  equal  to  the  sum  of  the 
quantities  of  work  performed  in  the  contrary  direction,  regarded  as 
negative.  In  other  words,  the  work  performed  by  the  entire  system 
of  impressed  and  inertia  forces,  taken  collectively,  must  be  zero. 

To  state  this  mathematically,  the  inertia  forces,  denoted  by  /t1  Iiy 
/j,  etc.,  are  exerted  in  and  by  the  elementary  masses,  m„  m2i  m3,  etc., 
respectively,  and  these  elementary  masses  describe  in  a  definite  time  t? 
the  respective  paths,  »„  «*  s3,  etc. 

Similarly,  the  points  of  application  of  the  impressed  forces,  whose 
intensities  are  denoted  by  Pl9  P2>  ^3>  etc.,  describe  in  the  definite 
time  t  paths  whose  lengths  are  denoted  by  pu  p&  p3,  etc. 

Let    &?„  6s2}  6s3,t  etc.,  denote  the  orthographic  projections  of   the 

paths  described,  in  the  element  of  time  dt,  by  the  points  of  application 

of  the  inertia  forces,  upon  the  respective    directions    of   those  forces. 

Then  will 

2  /.  6s  =  /j  cJ$j  -j-  /2  6s2  -f-  I3  5s3  -J-  etc., 

be  the  elementary  work  of  reaction  of  all  the  inertia  forces. 

And  if  6/>„  6p.2,  6p3j  etc.,  denote  the  orthographic  projections  of  the 
paths  described  by  the  points  of  application  of  the  extraneous  or 
impressed  forces,  on  the  respective  directions  of  those  forces,  during 
the  same  element  of  time  dt,  then  will  the  quantity  of  the  elementary 
work  of  the   impressed  forces  be 

lP6p=  P,  dPl  -f  P2  o>2  -f  P3  o>3  +  etc. 


36         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Regarding  the  algebraic  sum  of  the  work  of  the  impressed  forces 
as  positive,  and  that  of  the  inertia  forces  as  negative,  since  these  latter 
forces  oppose  all  changes  of  motion,  we  must  always  have 

2Po>  —  1,16s  =  0. 


Hut 


Hence, 


_.  dht      T  cPs*      T  d2So 

/1=m,  — ;  /^m,^;   /.  =  ~,^;   etc. 


2  Ids  =  2  m  —  6s; 
at4 


which  substituted  gives 

cPs 
2P6p—2m—ds  =  0      .     .     .     .     (A) 

And  this  is  the  single  formula  referred  to  in  the  preface  to  this 
book  as  the  one  fundamental  equation  which  embraces  in  its  dis- 
cussion the  whole  of  physical  and  mechanical  science. 

§  59.— For  the  sake  of  simplicity  in  the  demonstration,  we  have 
supposed  the  elementary  masses,  ?nu  m2,  m3,  etc.,  to  compose  a  single 
body.  But  it  is  evident  that  the  same  reasoning  is  applicable  to 
systems  of  bodies,  or  masses,  of  any  size,  connected  in  any  manner 
whatever;  such  as,  for  example,  machines  composed  of  many  parts; 
or  the  solar  system,  in  which  the  sun,  planets  and  satellites,  constantly 
pulling  each  other  together,  are  kept  from  falling  into  one  confused 
heap  of  ruins,  and  are  held  apart,  each  in  its  proper  orbit,  by  that 
precisely-balancing  resistance  which  the  action  of  gravitation  finds  ever 
opposed  to  it  in  the  exactly  equal  reaction  of  the  inertia  forces.  Than 
which  magnificent  example  of  perpetual  conservative  equilibrium,  noth- 
ing more  grand  is  known  to  us  in  the  material  world. 

§  60. — Our  Equation  (A)  is,  therefore,  perfectly  general,  or  appli- 
cable to  all  bodies,  or  systems  of  bodies,  connected  by  such  forces 
as  those  of  cohesion,  gravitation,  etc.,  or  in  any  manner. 

PRINCIPLE    OF    D'ALEMBERT. 

§  61. — The  forces  of  inertia  developed  by  the  impressed  force* 
P,  P\  P'\  &c,  may   or  may  not  be  equal  to  them,  depending  upon 


MECHANICS    OF    SOLIDS.  37 

the  manner  of  their  application.  If  the  impressed  forces  be  in  equi- 
librio,  for  instance,  they  will  develop  no  force  of  inertia;  but  in  all 
cases  the  forces  of  inertia  developed  will  be  equal  and  contrary  to 
so  much  of  the  impressed  forces  as  determines  the  change  of  motion. 
The  portions  of  the  impressed  forces  which  determine  a  change  of 
motion  are  called  effective  forces ;  and  from  Equation  (A)  we  infer 
that  the  impressed  and  effective  forces  are  always  in  equilibrio  when 
the  directions  of  the  latter  are  reversed.  This  is  usually  known  as 
D'Alemberfs  Principle,  and  is  nothing  more  than  a  plain  consequence 
of  the  law  that  action  and  reaction  are  ever  equal  and  contrary. 

This  same  principle  is  also  enunciated  in  another  way.  Since  the 
effective  forces  reversed  would  maintain  the  impressed  forces  in  equi- 
librio, and  prevent  them  from  producing  a  change  of  motion,  it  follows 
that  whatever  forces  may  be  lost  and  gained  must  be  in  equilibrio  ;  else 
a  motion  different  from  that  which  actually  takes  place  must  occur. 

VIRTUAL    VELOCITIES. 

§  62. — The    indefinitely    small   paths  mn,    m'n',    described   by  the 
points  of  application  of  the  forces  P  and  P'  during  the  slight  motion 
we  have  supposed,  are  called  virtual  veloci- 
ties ;    and  they  are  so  called,  because,  being 
the    actual    distances    passed    over   by   the 


points  to  which  the  forces  are  applied,  in         «, 

the    same   time,  they  measure   the   relative  r,~m'' —**J" 

rates  of  motion  of  these  points.     The  dis- 
tances rm  and  r'm',  represented  by  dp  and 

dp,'  are,  therefore,  the  orthographic  projections  of  the  virtual  velocities 
upon  the  directions  of  the  forces.  These  projections  may  fall  on 
the  side  toward  which  the  forces  tend  to  urge  these  points,  or  the 
reverse,  depending  upon  the  direction  of  the  motion  imparted  to 
the  system.  In  the  first  case  the  projections  are  regarded  as  positive, 
and  in  the  second  as  negative.  Thus,  in  the  case  taken  for  illustration, 
mr  is  positive  and  m'r'  negative.  The  products  P  dp  and  P'  dp  are 
called  virtual  moments.  They  are  the  elementary  quantities  of  work 
of  the  forces  P  and  P'.     The  forces  are  always  regarded  as  positive ; 


38         ELEMENTS    OF    ANALYTICAL   MECHANICS. 

the  sign  of  a  virtual  moment  will,  therefore,  depend  upon  that  of  the 
projection  of  the  virtual  velocity. 

§  63. — Referring  to  Equation  (A),  we  conclude,  therefore,  that 
whenever  several  forces  are  in  equilibrio,  the  algebraic  sum  of  their 
virtual  moments  is  equal  to  zero  ;  and  in  this  consists  what  is  called 
the  principle  of  virtual  velocities. 

That  this  is  true  is  evident,  for  if  the  impressed  forces,  P,  P\  P", 

etc.,  be  in  equilibrio,  they  will  develop  no  inertia,  and  Equation   (^4) 

will  reduce  to 

lP6p  =  0      ......      (26) 

Whatever  be  its  nature,  the  effect  of  a  force  will  be  the  same  if 
we  attribute  its  effort  to  attraction  between  its  point  of  application 
and  some  remote  point  assumed  arbitrarily  and  as  fixed  upon  its  line 
of  direction,  the  intensity  of  the  attraction  being  equal  to  that  of 
the  force.  Denote  the  distance  from  the  point  of  application  of  P 
to  that  toward  which  it  is  attracted  by  p,  and  the  corresponding 
distances  in  the  case  of  the  forces  P',  P",  &c,  by  p\  p",  <fec, 
respectively;  also,  let  dp,  dp',  dp",  <fcc,  represent  the  augmentation 
or  diminution  of  these  distances  caused  by  the  body's  motion,  supposed 
indefinitely  small;    then  Equation  (26)  may  be  written 

Pdp  +  P'dp'  +  P"dp"  +  &c.  =  0  .     .     .     (27) 

in  which  the  Greek  letter  6  simply  denotes  change  in  the  value  of 
the  letter  written  immediately  after  it,  this  change  arising  from  the 
small  displacement. 

§  64. — Conversely,  if  in  any  system  of  forces,  the  algebraic  sum 
of  the  virtual  moments  be  equal  to  zero,  the  forces  will  be  in  equi- 
librio. For,  if  they  be  not  in  equilibrio,  some,  if  not  all,  the  points 
of  application  will  have  a  motion.  Let  q,  q,  q",  <fec,  be  the  ortho- 
graphic projections  of  the  paths  which  these  points  describe  in  the 
first  instant  of  time,  and  Q,  Q',  Q",  &c,  the  intensities  of  such  forces 
as  will,  when  applied  to  these  points  in  a  direction  opposite  to  the 
actual  motions,  produce  an  equilibrium.  Then,  by  the  principle  of 
virtual  velocities,  we  shall  have 


MECHANICS    OF    SOLIDS.  39 

Pdp  -f  P'dp'  +  P"fy"  +  &C,  +   Qq+  Q'q'  +  Q"q"  +  &c.  ss  0. 
J iut  by  hypothesis, 

Pdp  +  P'o>'  +  P"o>"  +  <fcc.  =  0, 
and  hence, 

Qq  +  Q'q  +  C  V  +  &c  =  o   .    .    .    (28) 

Now,  the  forces  Q,  Q',  Q",  &c,  have  each  been  applied  in  a  direc- 
tion contrary  to  the  actual  motion ;  hence,  all  the  virtual  moments  in 
Equation  (28)  will  have  the  negative  sign ;  each  term  must,  therefore, 
be  equal  to  zero,  which  can  only  be  the  case  by  making  Q,  Q,'  Q", 
<fcc,  separately  equal  to  zero,  since  by  supposition  the  quantities 
denoted  by  q,  q',  q",  are  not  so.  AVe  therefore  conclude,  that  when 
the  algebraic  sum  of  the  virtual  moments  of  a  system  of  forces  is 
equal  to  zero,  the  forces  will  be  in  equilibrio. 

INTERPRETATION    OF    EQUATION    (A). 

§  G5. — -Clear  conceptions  of  the  meaning  of  each  of  the  terms  of 
Equation  (^1)  are  evidently  essential  to  the  further  understanding  of 
our  subject. 

Its  first  term  has  been  defined  to  be  the  elementary  work  of  the 
forces  P}  P',  P"t  etc.,  acting  in  the  directions  and  along  the  ele- 
mentary paths,  dp,  dp',  dp",  etc.,  and  upon  the  respective  masses, 
m,  m\  to",  etc.,  composing  the  body  or  system  of  bodies,  either 
directly  or  by  transmission  to  them. 

If,  therefore,  we  wish  to  obtain  the  finite  amount  of  work  done 
by  the  impressed  forces  during  an  interval  of  time,  in  which  they 
cause  the  system  to  change  from  one  position,  state,  or  configuration, 
(1),  to  another,  (2),  it  is  evident  that  we  must  integrate  the  first 
term  between  the  corresponding  limits  of  its  variables.  And  the 
definite  integral 


if  Pdp  =Q,-QX (29) 


will  express  the  amount  of  work  performed  during  the  change. 

For   such  a  dynamical    system  of   masses   and    forces   the    general 

integral  Q  =  If  Pdp (30) 

has  been  called  by  Gauss  and  others  the  potential.  Its  total  value 
would  evidently  be  that  obtained  by  integrating  between  the  limits 
of  p  equal  to  zero  and  p  equal  to  plus  and  minus  infinity.  But  for 
all  practical  cases  the  integration  can  only  be  between  definite  limits. 


40         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

To  make  this  clear  by  an  example,  take  the  descent  of  the  heavy 
weight  of  the  pile-driver,  which,  denoted  by  w,  falls  from  a  height, 
or  level,  h2,  to  a  lower  level,  hx,  then  will   (Equation  8)  the  work 

Q*  —  Qx  =  V>  (h2  —  hx) 
be  the  change  of   the  value  of    the  potential   for  such    a  fall  of   that 
weight. 

Take  again  the  descent  by  gravity,  or  fall,  of  a  given  weight  of 
water,  w,  for  a  mill  from  one  level,  or  dynamic  head,  hQ,  to  another, 
A„  then  the  same  equation, 

§2  —    Ql  =  W  (hi  ~  !h), 

expresses  the  amount  of  work,  or  change  of  the  potential,  due  to  the 
fall  of  the  given  mass.  And  it  is  so  called,  because  it  indicates  the 
availability  of  the  change  to  do  other  work,  such  as  driving  the  pile 
by  the  pile-driver,  or  grinding  corn,  sawing  wood,  etc.,  by  the  mill. 

Level  surfaces,  being  those  for  which  the  product  wh,  or  potential, 
has  the  same  value,  are,  therefore,  called  equipotential. 

§  66. — As  work  done  has  been  denoted  (in  Equation  *7)  by  the 
expression  Q  =  fPds, 

in  which  P  is  the  variable  force  exerted  to  overcome  a  resistance 
over  the  path  s ;  and  as  we  use  the  letters  P,  P',  P",  etc.,  to  denote 
the  impressed  forces  of  any  system,  the  integral 

Q  =  ZfPdp, 
whether  taken  alone,  or  used  as  in  Equation   (A),  may  be  ambiguous, 
or  capable  of  two  distinct  significations. 

It  may  be  taken  to  mean  either  the  kinetic  energy,  which  is  the 
work  done,  or  the  potential  energy,  which  determines  the  capacity  of 
the  forces  to  do  that  work,  and  these  two  energies  must  not  be  con- 
founded. This  important  distinction  can  be  made  clear  only  by  a 
thorough  understanding  of  terms  which  many  authors  too  often  use 
interchangeably,  although  representing  different  things. 

Then  when  we  speak  of  force  we  mean  the  essence  of  the  thing ; 
when  we  speak  of  intensity  we  mean  an  instantaneous  effort,  and  when 
of  the  measure  of  that  effort  we  mean  its  effect  exerted  during  a 
unit  of  time,  which,  as  we  have  seen,  is  represented  by  the  mass 
multiplied  by  the  velocity  of   a   free   body,  moved   from  rest,  due  to 


MECHANICS    OF    SOLIDS.  41 

that  effort.  "When  we  speak  of  work  we  mean  that  measure  repeated 
as  many  times  as  the  path  described,  while  the  effort  acts  and  is 
supposed  constant,  contains  the  unit  of  path. 

Therefore  the  measure  of  the  force's  intensity  is,  in  fact,  work, 
hut  a  definite  quantity  of  work,  and  is  the  work  of  that  force  over 
a  unit  of  space. 

We  know  that  the  effect  of  a  force  is  very  different  acting  under 
different  circumstances.     If  acting  on  a  free  body  at  rest  it  will  pro- 

MV2 

dncc    a  quantity  of   work  represented    by  — — .     But  we   know  that 

MV2 

the  work   embodied   in    may   be   transformed    into   some    other 

representative  of  work,  and  since  it  may  thus  be  transformed,  this 
quantity  of  work  is  called  kinetic  energy,  because  it  involves  the  idea 
of  motion. 

If  the  force  act  to  bend  a  spring,  for  example,  or  transform  the 
shape  of  a  body,  the  latter  when  relieved  from  the  action  of  the 
force  will  restore  the  work  communicated  to  it,  and  the  whole  measure 
of  its  capacity  to  do  so  is  called  potential  energy. 

To  illustrate,  if  we  suppose  muscular  effort  to  be  employed  in 
bending  a  bow,  to  which  an  arrow  is  adjusted,  this  is  an  instance  of 
potential  energy  being  stored  in  the  changed  elasticities  of  the  fibres 
of  the  bow,  and  when  these  are  allowed  to  act  on  the  arrow,  through 
the  intervention  of  the  cord,  the  potential  energy  is  very  quickly 
being  transformed  into  kinetic  energy  of  the  arrow,  which  attains  its 
maximum  when  the  arrow  leaves  the  cord. 

The  arrow  carries  this  kinetic  energy  with  it  through  space,  and  if 
the  arrow  in  its  flight  meet  another  bow-string,  it  will  deliver  up  to 
this  bow  its  kinetic  energy,  being  transformed  into  an  equivalent 
amount  of  potential  energy,  and  become  stored  up  in  increased  elas- 
ticities of  the  fibres  of  the  second  bow.  When  all  is  thus  transformed, 
the  latter  gives  back  its  accumulated  potential  energy,  and  the  arrow 
leaves  it  with  its  restored  kinetic  energy,  and  thus  there  would  be 
forever  a  mutual  equal  interchange  of  these  two  energies. 

When,  therefore,  in  the  following  pages  the  term  energy  is  employed, 
it  must  be  understood  to  mean  accumulation  of  intensity,  frequently, 
though  very  improperly,  called  quantity  of   force,  the  terms  potential 


42         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and  kinetic  being  simply  adjectives  to  qualify  the  nature  of  this 
accumulation. 

§  67. — It  thus  appears  that  the  modern  and  expressive  name  of 
kinetic  energy  may  be  applied  to  denote  the  ability  of  a  moving  mass 
to  perform  work  by  its  inertia;  and  that  this  term  may  be  very 
advantageously  substituted  for  that  of  the  half  of  the  vis  viva,  or  half 
of  the  living  force,  which  are  only  unmeaning  names  for  the  algebraic 
quantity  ^*  =  fp^ 

23 

which  we  have  shown  (in  Equation  21)  measures  the  quantity  of 
work  of  the  inertia  forces  of  a  body  in  acquiring,  or  losing,  its 
velocity. 

§  68. — All  the  various  forms  in  which  the  physical  forces  of 
nature  present  themselves,  as  the  causes  of  observed  effects,  may  now 
be  simply  classified  under  the  two  general  heads  of  'potential  and 
kinetic  energy. 

The  forces  which  act  at  sensible  and  measurable  distances,  in  the 
phenomena  of  gravitation,  electrical  and  magnetic  induction,  attraction 
and  repulsion ;  or  those  more  hidden  forces  which  act  only  at  insensi- 
ble distances  to  combine  atoms  into  molecules  and  molecules  into 
masses,  or  which  at  times  separate  molecules,  as  in  the  phenomena 
of  elasticity,  or  those  of  chemical  and  electrolytic  decomposition ; 
these  all  are  but  instances  of  potential  energy  performing  work  by 
changes  of  position. 

On  the  other  hand,  the  phenomena  of  the  winds  due  to  solar 
action,  the  radiant  heat  and  light,  forms  of  kinetic  vibratory  energy, 
emanating  from  the  sun  and  burning  bodies,  the  inertia  of  projectiles 
and  other  moving  masses  when  their  motions  are  changed,  these  are 
instances  of  what  was  formerly  called  vis  viva,  but  is  now  better 
named  kinetic  energy. 

§  69. — It  has  been  stated  that  energy,  though  readily  transmutable 
from  one  of  its  forms  into  another,  is  not  destructible.  Thus  heat  is 
changed  into  mechanical  work"  in  the  steam-engine,  and  mechanical 
friction  develops  or  rather  is  changed  into  heat.  Chemical  affinity, 
by  union  of  zinc  with  oxygen,  develops  electrolytic  action,  which  may 


MECHANICS    OF    SOLIDS.  43 

be  made  to  decompose  oxide  of  zinc,  or  drive  an  electro-magnetic 
engine  doing  mechanical  work.  And  this  electrical  force  may  be 
transmitted  through  a  metallic  wire  to  long  distances ;  a  fact  first 
discovered  about  a  century  and  a  half  ago  by  Stephen  Gray,  an 
invalid  pensioner  in  a  charity  hospital,  experimenting  to  relieve  the 
tedium  of  sad  hours,  and  little  dreaming  of  the  future  importance 
his  discovery  would  acquire  when  used  to  convey  swift  messages  of 
human  affairs  around  our  globe. 

Many  and  beautiful  indeed  are  the  transformations  of  energy; 
marvelous  also  are  the  changes  due  to  the  physical  forces  and  the 
mutual  dependence  of  terrestrial  phenomena  and  of  vegetable  and 
animal  life  upon  each  other  and  upon  solar  action. 

Certainly,  too,  nothing  can  be  better  calculated  to  cause  us  to 
form  proper  conceptions  of  the  infinite  power,  wisdom,  and  goodness 
of  the  Divine  Ruler  of  the  Universe,  than  the  study  and  contempla- 
tion of  the  simple  laws  ordained  for  its  continued  harmonious  exist- 
ence amid  perpetual  change,  yet  all  without  the  least  destruction  of 
force  or  matter. 

These  subjects  are  full  of  varied  interest,  but  they  do  not  belong 
to  Analytical  Mechanics,  except  in  so  far  as  their  discussion  may  be 
requisite  to  set  forth  the  full  meaning  and  scope  of  the  laws  of  force 
and  motion,  and  to  give  proper  conceptions  of  their  relations  to  other 
branches  of  physical  science. 

§  70. — It  is  now  evident  that  energy  may  be  given  to  and  become 
stored  up  in  dead  masses,  either  as  potential,  or  as  kinetic,  energy ; 
and  in  either  case  it  is  capable  only  of  passing  from  one  of  these 
forms  into  the  other,  without  loss  or  gain.  Nothing  can  be  created, 
nothing  destroyed,  except  by  God  himself. 

The  dormant  potential  energy  of  gunpowder,  brought  into  action 
by  a  spark,  is  simply  transferred  as  kinetic  energy  to  the  ball  which 
tears  apart  the  particles  of  a  mass. 

The  muscular  effort  which  expends  itself  in  winding  up  the  weight 
or  spring  of  a  clock  is  only  converted  into  potential  energy  of  grav- 
itation, or  elasticity,  to  be  in  turn  transformed  into  actual,  or  kinetic, 
energy  when  it  gives  motion  to  the  wheelwork.  Those  wheels  also 
transfer  their  kinetic  energy  to  adjacent  particles  of  air  and  the  sup 


44         ELEMENTS    OF   ANALYTICAL    MECHANICS. 

porting  framework,  to  be  in  turn  indefinitely  given  to  other  surround- 
ing portions  of  matter.  Such  is  the  conception  of  what  is  called 
conservation  of  energy. 

§  71. — We  end  this  discussion  by  simply  drawing  attention  to  the 
fact  that  as  the  first  term  of  Equation  (^1)  is  susceptible  of  two 
different  meanings,  related  as  cause  and  effect,  so  also  has  the  second 
term  two  distinct  significations. 

Inertia  being,  by  definition,  resistance  to  any  change,  whether  of 
acceleration  or  retardation,  it  is  evident  that,  when  a  system  is 
acquiring  motion,  the  second  term  of  Equation  (A),  or  rather  its 
definite  integral  P2    d'zs  m 


will  indicate  work  of  inertia  done,  by  expenditure  of  potential  energy 
converted  into  gain  of  kinetic  energy.  While,  on  the  contrary,  when 
the  system  is  retarded,  the  same  integral  must  express  loss,  or  expend- 
iture of  kinetic  energy  in  work  done,  or  potential  energy  stored. 

REFERENCE    TO    CO-ORDINATE    AXES. 
§  72. — First    Transformation.     Equation    (A)    is    of   a    form   too 
general   for  easy  discussion,  and    may  be    simplified   by  referring   the 
forces  and  motions  to  rectangular  axes. 

Denote  by  a,  /3,  y,  the  angles  which  the  direction  of  the  force 
P  makes  with  the  axes  .?,  y,  z,  respectively ;  by  a,  b,  c,  the  angles 
which  its  virtual  velocity  makes  with  the  same  axes ;  and  by  </>,  the 
angle  which  the  virtual  velocity  and  direction  of  the  force  make  with 
each  other,  then  will 

cos  (j)  =  cos  a  .  cos  a  -f  cos  b  .  cos  (5  -\-  cos  c  .  cos  y. 
Denote    by  k   the  virtual    velocity,  and    multiply  the   above    equation 
by  Pic,  and  we  have 

Pk  cos  <f>  ■=  Pk  cos  a  .  cos  a  -f-  Ph  cos  b  .  cos  j3  -f-  Pk  cos  c  .  cos  y. 
But  denoting   the    co-ordinates   of  the  point  of   application    of   P  by 
~>  y,  z,  we  have 

k  cos  (f>=:  dp;   k  cos  a  =  6x ;    k  cos  b  =  dy;    k  cos  c  =  8z ; 
end  these  values  substituted  above,  give 

P  .  dp  =  P  cos  a  .  6x  -\-  P  cos  (3  .  8y  +  P  cos  y  .  6z.     .     (31 ). 
Similar  values  mav  be  found  for  the  virtual  moments  of  other  forces. 


MECHANICS     OF    SOLIDS.  45 

§  73. — If  P  be  replaced  by  the  force  of  inertia,  then  will  a,  (3,  and 
y  denote  the  inclinations  of  the  direction  of  this  force  to  the  axes  xyz: 
k.  its  virtual  velocity  ;  «,  6,  and  c  tlie  inclinations  of  the  latter  to  tire* 
axes,  and  (ji  its  inclination  to  the  direction  of  the  force  of  inertia,  and 
we  may,  Eq.  (13),  write 

d2g  i  <***  i  d°'*       a  i       i.         du  i 

m  •  — -  k  cos  <b  =  m  -— -  •  cos  a .  k  cos  a  +  w  — -  cos  j» .  k  cos  b  -4-  m  ——  cos  y .  k  eoa  c 

rf/a  v  d£*  T      dp  ^      dP        f 

But 

&  cos  §  ■=.  §s\      k  cos  a  =  6x ;       &  cos  b  =  dy\       k  cos  e  =n  $2 ; 

g?2s.cos  a  =  <Px\     d2s  cos  j3  =  cT2y;     rf's  cos  y  =  c?2z ; 


whence, 


ra 


d*s     t  (Px  d?y     x  d2z 


and  similar  expressions  may  be  found  for  the  virtual  moments  of  the 
forces  of  inertia  of  the  other  elementary  masses. 

§  74. — If  the  intensity  of  the  force  P,  be  represented  by  a  portion 
of  its  line  of  direction,  which  is  the  practice  in  all  geometrical 
illustrations  of  Mechanics,  the  factors  P  cos  a,  P  cos  /3,  and  P  cos  y, 
in  Equation  (31),  would  represent  the  intensities  of  forces  equal  to 
fhe  projections  of  the  intensity  P,  on  the  axes ;  and  regarding  these 
as  acting  in  the  directions:  of  the  axes,  the  factors  dx,  Sy,  and  8z,  will 
represent  the  projections  of  their  virtual  velocities,  which  virtual  veloci- 
ties will  coincide  with  that  of  the  force  P. 
Again,  Equation  (32), 

d2x  d2y  d2z 

m-~d?  m'w  m-ir*' 

are  forces  of  inertia  in  the  directions  of  the  axes,  and  Sx,  Sy,  Sz,  are 
the  projections  of  their  virtual  velocities ;  these  virtual  velocities  coincide 
with  that  of  the  inertia  of  m. 

The    values  of   these  virtual    velocities   depend   upon   the  nature  of 
the  displacement. 


46 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


FREE    MOTION    OF    A    RIGID    SYSTEM. 


§  75. — Second  Transformation.     By  the  substitution,  in  Equation  {A). 

d*  s 
for  P6p  and  m .  j-j  .  ds,  their  values  in  Equations  (31)  and  (32),  there 

would  result  an  equation  containing,  in  general,  three  times  as  many  vari- 
ations of  xyz  as  there  are  extraneous  forces  and  elementary  masses,  m. 
Where  the  forces  are  applied  to  a  body  whose  elementary  masses  are  in- 
variably connected — that  is,  to  a  rigid  solid — the  number  of  these  varia- 
tions is  greatly  reduced,  in  consequence  of  the  relations  determined  by 
this  connection. 

The  most  general  motion  we  can  attribute  to  a  body  is  one 
of*  translation  and  of  rotation  combined.  A  motion  of  transla- 
tion carries  a  body  from  place  to  place  through  space,  and  its 
jH)sition,  at  any  instant,  is  determined  by  that  of  some  one  of  its 
elements.  A  motion  of  rotation  carries  the  elements  of  a  body 
around  some  assumed 
point.  In  this  investi- 
gation, let  this  point 
be  that  which  deter- 
mines the  body's  place. 

Denote  its  coordi- 
nates by  xt  yt  zt  and 
those  of  the  element 
*w,  referred  to  this  point 
as  an  origin  by  x',  y\ 
z' ;  there  will  thus  be 
two  s^ts  of  axes,  and 
supposing  them  parallel, 
we   have 


and  differentiating. 


x  =  xi  -{-  %', 

y  =  y,  +  y', 

z  =  zt  +  z'; 
dx  =  dx,  -\-  dx', 

dv  =  dy,  +  *y\ 

dz  =  dzt  ■+■  dz'. 


(33), 


f 


(34). 


MECHANICS    OF     SOLIDS. 


47 


Demit  from  m  the  per- 
pendiculars mX\  m  Y ',  mZ\ 
upon  the  movable  axes. 
Denote  the  first  by  r\  the 
second  by  r",  and  the  third 
by  r"\  Let  0\  0",  0"', 
be  the  projections  of  m  on 
the  planes  xy,  xz,  yz,  re- 
spectively. Join  the  several 
points  by  right  lines  as 
indicated  in  the  figure. 

Denote  the  angle 


Then  will 


X'CO'     by  p, 

Y'CO'"  by  *, 
Z'CO"     by  i/>. 

x'  =  r'"  cos  <p, 


r'"  sin  (f>, 


the  triangle  X'CO'  give    \    , 

t 

the  triangle  Z'CO".  \    . 

6  (  z '  = 

the  triangle  F'CO"',  \    ,         ,     .       't 

°  (  2  =  r    sin  sr,  f 


#'  =V  sin  ip,  -I 
r"  cosi/?,  [ 


(35), 


(36), 


(37). 


We  here  have  two  values  of  x\  one  dependent  upon  0,  and  the 
other  upon  tp.  If  the  body  be  turned  through  an  indefinitely  small 
angle  about  the  axis  z\  the  corresponding  increment  of  x'  is  obtained 
by  differentiating  the  first  of  Equations  (35) ;    and  we  have 

dx'  =  —  /"  sin  $ .  d(ff ; 

if  it  be  turned  through  a  like  angle  about  the  axis  y\  the  correspond- 
ing increment  of  x'  is  found  by  differentiating  the  first  of  Equations 

(36),  and 

dx'  =  r"  cos  i/> .  dtp. 

If  these  motions  take  place  simultaneously  about  both  axes,  the  above 
become  partial  differentials  of  x\  and  we  have  for  its  total  differential, 


dx'  =  r"  cos  ip .  dip  —  r"  sin  <p  .  d(j>) 


48 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


replacing  r"  cos  4*  and  r"'  sin  9,   by  their   values  in  the.  above  Equa- 
tions, and  we  get 

dx'  =  z'.d-^  —  y'.c?(p; 

and  in  the  same  way, 

dy'  =  x' .  dcp  —  zf .  d  &, 
dz'  =  y'  .dtx  —  x' .  d  4, 

which  substituted  in  Equations  (34),  give 


(38) 


dx  =  dx4  -\-  zf  .d  \  —  y 
dy  =  dy4  -f-  x'  .d  <p  —  z 
dz  =  d  zt  4-  y'  >d  &  —  x'.  d 4. 


.(39) 


and  because  the  displacement  is  indefinitely   small,  we  may  write 


dx  =  8xt  +  zr  .8-^  —  y'  .  #<p, 
8y  =  Syi-\-xf.dcp  —  zf .  Six, 
Sz  =  S  zt  +  y' .  S  tx  —  x' .  S  4 ; 

and  tnese  in  Equations  (31)  and  (32),  give 

"  P  cos  a.  8xt  -\-P  cos  /3 . Syt  -\- P  cos  7 .  Szt 
p    *  \    -\-  P  .  (xr .  cos  (3  —  y' .  cos  a) .  &p 

+  P .  (zf .  cos  a  —  #' .  cos  y)  .  S-^ 
k  +  P  ,{y' .  cos  y  —  zr .  cos  /3) .  fe. 


.  (39)' 


d*2 


.&=•{ 


<ft2        '    '  dt2 

x' .  d?y  —  y' .  d2x 


(Pz    , 


-f-  «*• 

4-  m 
4-  w* 


dP 
zf .  cPa:  —  a;' .  c?22 

y' .  cPz  —  2' .  c?2y 
~dl2 


(Ps' 


Similar  values  may  be  found  for  P' .  Spf  and  m' .  — —  »  5s'  &c.     in 

a/1 

these  values  ##,,  § yt     and  £z/?  will  be  the  same,  as  also  Sep,  5 4,   and 

<$-#,   for   the   first  relate   to  the    movable    origin,  and  the  latter  to  the 

angular   rotation    which,   s'nee   (he    body    is    a  solid,   must   be   of  oqual 


MECHANICS     OF     SOLIDS. 


49 


values  for  all  the  elements;  so  that  to  find  the  values  of  the  virtual 
moments  of  the  other  forces,  it  will  be  only  necessary  suitably  to 
accent  P,   a,  /3,  7,    *,  y,  z,    x\  y',  z\ 

These    values   being   found   and   substituted   in   Equation   (A),   we 
shall  find, 

d2x 


/  d2x\ 

(  2  P .  cos  a  —  2»i-  -— - 1  $x4 

+  (2P.cos/3-  Sf»-^)  fy 

/  d2z  \ 

+  (2P.cosy-2m~J^ 


a;' .  c?2y  — 


-  y' .  d2xl 

* — J 


dt2 

z' .  d2x  —  x  .  d2z 


-f-  2  P.  [x'.  cos/3—  y'.  cos  a)— 2  m,' 
-h  I  2  P.  (z' .  cos  a  —  x.  cos  7)  —2  m 
4-   [ 2  P.  (y\  cos  7-2'.  cos  /3)-2  m  •  *' -^  -  z\.<Py\  g 


J  9 


^0.(40) 


J  <ty 


tf 


Now  the  displacement  was  wholly  arbitrary;  the  v  a  Lies  of  dxt,  dyi% 
dzt,  6  p,  (5  J,,  d&,  determined  by  this  displacement,  are  also  arbitrary; 
whence,  by  the  principle  of  indeterminate  co-efficients, 


2  P .  cos  a  —  2  m  •  — —■  =  0, 

dt2 

2  P .  cos  /3  -2m-  -,2f-  =  0,     ), 

c«2 


(-B) 


d2z 
2  P . cos  7  —  2m.  — —  =  0 ; 
1  dt1 


J 


2  P .  (xf .  cos  /3  —  y' .  cos  a)  —  2  ;w  • 
2  P .  (s' .  cos  a  —  x  .  cos  7)  —  2  7«  • 
2  P  .  (y' .  cos  9  —  2'  .  cos  /3)  —  2  m  • 


x'.  d2y  —  y'.  d2x 

It2 
z'.  d2x  -  x'.  d2z 
d(* 

y'.  (Fz  —  z'.  d2y 


=  0, 
=  0, 
=  0. 


.   (C) 


50  ELEMENTS    OF     ANALYTICAL    MECHANICS. 

1  §76. — These  six  equations  express  either  all  the  circumstances  of 
motion  attending  the  action  of  forces,  or  all  the  circumstances  of 
equilibrium  of  the  forces,  according  as  inertia  is  or  is  not  brought 
into  action ;  and  the  study  of  the  principles  of  Mechanics  is  little 
else  than  an  attentive  consideration  of  the  conclusions  which  follow 
from   their  discussion; 

Equations  (B)  relate  to  a  motion  of  translation,  and  Equations 
(0)  to  a  motion  of  rotation.  They  are  perfectly  symmetrical  and 
may  be  memorized  with  great  ease. 

COMPOSITION     AND     RESOLUTION     OF    FORCES. 

§  77. — When  a  free  body  is  subjected  to  the  simultaneous  action 
of  several  extraneous  forces  which  are  not  in  cquilibrio,  its  state  will 
be  changed ;  and  if  this  change  may  be  produced  by  the  action  ol 
a  single  force,  this  force  is  called  the  remltant,  and  the  several  forces 
are  termed  components. 

The  resultant  of  several  forces  is  a  single  force  which,  acting  alone, 
will  produce  the  same  effect  as  the  several  forces  acting  simultaneously ; 
and  the  components  of  a  single  force,  are  several  forces  whose  simnhn- 
neous  action  produces    the   same    effect   as    the   single  force. 

If,  then,  several  extraneous  forces  applied  to  a  body,  be  not  in 
equilibrio,  but  have  a  resultant,  a  single  force,  equal  in  intensity  to 
this  resultant,  and  applied  so  as  to  be  immediately  opposed  to  it, 
will  produce  an  equilibrium;  or,  what  amounts  to  the  same  thing, 
if  in  any  system  of  extraneous  forces  in  equilibrio,  the  resultant  of  all 
the  forces  but  one  be  found,  this  resultant  will  be  equal  in  intensity 
and  immediately  opposed  to  the  remaining  force  ;  otherwise  the  sys- 
tem could   not   be   in  equilibrio. 

Conceive  a  system  of  extraneous  forces,  not  in  equilibrio,  and 
applied  to  a  solid  body,  and  suppose  that  the  equilibrium  may  be 
pioduced  by  the  introduction  of  an  additional  extraneous  force. 
Denote  the  intensity  of  this  force  by  i?,  the  angles  which  its  direc- 
tion makes  with  the  axes  z,  y  and  2,  by  a,  b  and  c,  respectively, 
and  the  co-ordinates  of  its  point  of  application  by  .r,  y,  z.  Then, 
because  the  inertia  cannot  act,  d2x,  d2y,  d2z  will  be   ztro,  and  taking 


MECHANICS     OF    SOLIDS. 


51 


the   two   origins   to   coincide,  Equations  (P)  and  (C),  will  give 

R  cos  a  +  P*  cos  a'  -f-  P"  cos  a"  +  P'"  cos  a'"  +  &c.  =  0, 

R  cos  b   4  P'  cos  /3'  4  P"  cos  /3"  +  P'"  cos  /3'"  +  &e.  =  0, 
£  cos  c   +  P'  cos  7'  4  P"  cos  7"  -f  P'"  cos  7'"  4.  &c.  =  0  ; 


it  (#  cos  b  —  y  cos  a)  -f-  P'  {%'  cos  /8-  —  y'  cos  a') 
4  P"  (x"  cos  £"  -  y"  cos  a")  +  &c. 

P  (2  cos  a  —  £  cos  c)   4  /"  (2'  cos  a'  —  x'  cos  7') 
4-  P"  (2"  cos  a"  —  x"  cos  7")  4  &c. 

P  (y  cos  c  —  z  cos  6)  4  /**  (yf  cos  7'  —  2'  cos  /3') 
4  P"  (y"  cos  7"  -  z"  cos  £")  4-  &c. 


[=0. 


Now  P  is  equal  in  intensity  to  the  resultant  of  all  the  othei 
forces  of  the  system,  or  in  other  words,  to  the  resultant  of  all  the 
original  forces ;  and  if  we  give  it  a  direction  directly  opposite  to 
that  in  which  it  is  supposed  to  act  in  the  above  equations,  it  be- 
comes in  all  respects  the  same  as  that  resultant,  being  equal  to  it 
in  intensity  and  having  the  same  point  of  application  and  line  of 
direction.  Adding,  therefore,  180°  to  each  of  the  angles  a,  6,  and  c, 
the  first  terms  of  the  foregoing  equations  become  negative,  and 
transposing  the  other  terms  to  the  second  member  and  changing  all 
the   signs,  we   have, 


R  cos  a-P'  cos  a'  +  P"  cos  a"  4  P'"  cos  a'"  4  &c.  =  X; } 
R  cos  b  =  P'  cos  0'  4-  P"  cos  $"  4  P"'  cos  &'"  4  &c  =  Y\  \ 
R  cos  c  =  P'  cos  7'  4  P"  cos  7"  4  P'"  cos  7'"  4  &c.  =  Z.  J 


(41) 


R  (x  cos  I  —  y  cos  a)  =  1 


P'  (x'  cos  /5'  —  y'  cos  a') 

4P"(*"cos/3"-y"cosa")  )>=Z; 
4  &c. 


P'  (2'  cos  a'  —  a;'  cos  7') 
R  (z  cos  a  —  x  cos  c)  =  ^  4?"  (z"  cos  a"  —  x"  cos  7") 

f  P'  (y'  cos  7'  -  z'  cos  /3') 
ft  (y  cos  c  —  2  cos  b)  =  <  +P"  (y"  cos  7"  —  z"  cos  ,8") 


=Jf; 


V  •  (42) 


=M 


52  ELEMENTS    OF    ANALYTICAL     MECHANICS. 

or, 

R  cos  a  =  X, > 
R  cos  6  =  y, 

72  COS  C  =z    Z. 


(43) 


i2  (a:  cos  b  —  y  cos  a)  =  Z, 

i2  ( z  cos  a  —  #  cos  c)  =  iV,   > (44) 

R  (y  cos  c  —  z  cos  b)  —  N.  ^ 

Eliminating  ii!  cos  a,  R  cos  b  and  72  cos  c,  from  Equations  (44), 
Dy  means  of  Equations  (43),  we  get,  by  transposing  all  the  terms  to 
the  first  member, 

Xy  -  Yx  +  L   =  0,  ] 

Zz  -  Xz  -f  >/  =  0,   \ •     (45) 

Yz  -  Zy  +  N  =  0.  J 

Either  one  of  these  equations  is  but  a  consequence  of  the  other  two. 
They  are,  therefore,  the  equations  of  a  right  line — the  locus  of  tlte 
points  of  application ;  and  from  which  it  is  apparent,  that  the  point  of 
application  of  a  force  may  be  taken  anywhere  on  its  line  of  direction, 
.within  the  limits  of  the  body,  without  altering  the  etfects  of  the  force. 
The  condition  expressive  of  the  existence  of  the  dependence  of  one  of 
these  equations  on  the  others,  will,  also,  express  the  existence  of  a  single 
resultant. 

§7S. — To  find  this  condition,  multiply  the  first  of  these  Equations 
by  Z,  the   second   by    Y,    the   third   by    X,    and   add    the    products 
we   obtain, 

ZZ+  F¥+  JJ=0 (46). 

§79. — Having  ascertained,  by  the  verification  of  this  Equation, 
that  the  forces  have  a  single  resultant,  its  intensity,  direction,  and 
the  equations  of  its  direction  may  be  readily  found  from  Equations 
(43)  and  (44). 

Squaring   each  of  the   group  (43),  and   adding,  we   obtain, 

R2  (cos2  a  4-  cos2  b  +  cos2  c)  =  X2  +■   Y2  +  Z2. 


MECHANICS    OF    SOLIDS, 


53 


Extracting    the    square   root    and   reducing   by  the   relation, 


cos2  a  -f-  cos2  b  -f  cos2  c  =  1, 


there  will  result, 


R  =  V X*  +  Y*  +  Z* 


(47) 


which    gives   the   intensity   of  the   resultant,    since   X,    Y  and  Z  are 
known. 

Again,  from  the  same  Equations, 


cos  a  =  — 


5 


cos  b  = 


cos  c  =  — - 


R 

Y 

R 

Z 

R' 


which   make   known    the   direction  of  the   resultant. 
The   group  of  Equations  (45)  give, 

Xy  -  Yx  +  XP'  (cos/S's'  -cos  a'/)  =  0,  j 
Zz-Xz  +  ZR'  (cos  a'  2'  -  cos7'x')  =  0,   J>    . 
Y z  -  Z  y  +  2  />'  (cos  7'  /  -  cos  /6  V)  =  0.  ] 

which   are    the   equations  of  the   line  of  the  resultant. 


(48) 


(49) 


PARALLELOGRAM    OF   FORCES. 


§80.  —If  all   the  forces   be   applied   to  the   same   point,  this  point 
may  be   taken  as   the   origin  of  co-ordinates,  in  which  case, 

x'  =  x"  =  x'"  &c  ==  0, 
y'  =  y"  =  y"'  &c.  =  0. 
z'  =  z"  =  z'"  &c.  =  0, 

and   the   last    term  in    each  of  Equations    (49),  will    reduce    to   zero. 
Hence,   to    determine    the    intensity     direction    and    equations    of    th« 

5 


54 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


line    of    direction    of    the   resultant,    we    have,    Equations    (47),    (48) 
and  (49), 

R  ==  y/X*  +P  +  22    ....         (50) 


X    \ 
cos  a  =  —  j 

R 

,      r 

cos  6  =  -—•> 
R 


cos  c  =  —  > 


(51) 


Xy  -  Fa:  =  0, 

Zx  -  Xz  =  0,  [ (52) 

•'      Yz  -  Zy  =  0.} 

The  last  three  equations  show  that  the  direction  of  the  resultant 
passes  through  the  common  point'  of  application  of  all  the  forces, 
which    might   have   been    anticipated. 

§81. — Let  the  forces  be  now  reduced  to  two,  and  take  the  plane 
of  these  forces  as   that  of  x  y ;    then  will 

/  =  y"  =  y'"  =  &c.  =  90°  ;  z  =  0, 

the   last   Equation  of  group  (41)  reduces   to, 

Z  =  0; 

and   the   above  Equations   become, 

R  =  ^/X2  +  Y2 (53) 


X    ~}. 


cos  a  =  —  ? 
R 

,        Y 

COS  0    J-z  —■> 

R 


(54) 


cos  c  =  0, 

Xy  -  Yx  =  0 (55) 

The  last  is  an  equation  of  a  right  line  passing  through  the 
origin.  The  direction  of  the  resultant  wilt,  therefore,  pass  through  the 
point  of  application  of  the  forces.  The  cos  c  being  zero,  c  is  90°, 
and   the  direction  of  the  resnl'ant  is  therefore  in  the  plane  of  the  forces. 


MECHANICS     OF     SOLIDS. 


55 


Substituting  U)    Equation    (53),    for  X.  and    K,    their    values    from 
Equations  (41),  we  obtain. 

R  =  y/  {Pf  cos  a'  +  P"  cos  a")2  +  (P~cos  /3'  +  P"  cos  /3")2  ; 
and  since 


cos2  a'    -|-  cos2  ,6'    =1, 
cos2  a"  -f  cos2  £"  t=  1, 

this    reduces   to 


m 


R  -  y/P"1  +  P '2  +  2  P'  P"  (cos  a'  cos  a"~4-  cos  /3'  cos  /3") ; 

denoting    the   angle    made    bv    the    directions  of*  the  forces    by  8,   we 

have, 

cos  a'  cos  a"  -f-  cos  /3'  cos  /3"   =  cos  5  ; 


and  therefore, 


i?  =  y/P'*   +  P"2  4-  2  P'  P"  cos  5 


(56) 


from  which  we  conclude  that  the  intensity  of  the  resultant  is  equal 
to  that  diagonal  of  a  parallelogram  whose  adjacent  sides  represent  the 
directions  and  intensities  of  the  components,  which  passes  through  th* 
point  of  application. 

§82.— Substituting  in  Equations  (54),  the  values  of  X  and  Y,  from 
Equations  (41),  we  have, 

R  cos  a  =  P'  cos  a'  4-  P"  cos  a", 
R  cos  b  =  P'  cos  /3'  4-  P"  cos  /3", 
and   because 

a'   =  90°  -  /S', 
a"  =  90°  -  /3", 
a     =  90°  -  6, 

these    Equations   reduce   to, 

i2  cos  a  =  P'  cos  a'  4  P'f  cos  a", 
#  sin  a  =  P'  sin  a'  +  P"  sin  a"  ; 


66  ELEMENTS    OF    ANALYTICAL    MECHANICS, 

and,  by  division, 

•  Ttl        '  >       i         Till        '  II 

sin  a      P  sin  a  -f-  P    sin  a 

• 

eosu  "  P'  cos  a'  +  P"  cos  a'7 ' 
clearing  fractions  and  transposing,  we  find, 

P"  (sin  a"  cos  a  —  cos  a"  sin  a)  ==  ■/*'  (sin  a  cos  a'  —  cos  a  sin  a') ; 

whence, 

P'        sin  a"  cos  a  —  cos  a"  sin  r/.       sin  (a"  —  a)  i~tt\* 

P"         sin  a  cos  a'  —  cos  a  sin  a'         sin  (a  —  a') 

That  is  to  say,  the  intensities  of  the  components  are  inversely  propor- 
tional to  the  sines  of  the  angles  which  their  directions  make  with  that 
of  their  resultant ;  but  this  is  the  relation  that  subsists  between  the 
two  adjacent  sides  of  a  parallelogram  and  the  sines  of  the  angles  which 
they  make  with  the  diagonal  through  their  point  of  meeting.  Whence, 
Eqs.  (06)  and  (5«)\ 

The  resultant  of  any  two  forces,  applied  to  the  same  point,  is  repre- 
sented, in  intensity  and  direction,  by  that  diagonal  of  a  parallelogram  of 
which  the  adjacent  sides  represent  the  components. 

Making 

a  —  a'  =  the  angle  R  m  P'  =  p' 

and 

R  +  P>  +  p> 

—  o, 

2 

we  have,  from  the  usual  trigonometrical  formula, 


^'=\^HrJ^ W 


§  83. — In  the  triangle  R  m  P',  since  P'  R  is  equal  and  parallel  to 
the  line  which  represents  the  force  P",  the  angle  m  P' R  =  <p,  is  the 
supplement  of  the  angle  d,  made  by  the  directions  of  the  components, 
and  there  will  result  the  following  equation  : 


.u        .    .  /{S  -  P')  (S  -  P") 


(58) 


MECHANICS     OF    SOLIDS. 


57 


Equation  (57),  will  make  known  the  angle  made  by  the  direction 
of  the  resultant  with  that  of  either  of  two  oblique  components,  pro 
vided,  the  intensities  of  the  components  and  resultant  be  known. 

§S4. — Also,  from  the  two  triangles  RmP:   and    RmPf\    we  find, 
.       .       F" .  sin  S 


sm  <p    = 


sin  <p 


tr 


R 

F'  .  sin  5 
~R~ 


►  '  '  (5»), 


■m 


from  which  the  angles  made 
by  the  direction  of  the  result- 
ant with  its  two  components  may  be  found. 

§85. — Let  there  now  be  the  three    forces    P,   P\    P",    applied    to 
the     material     point     w,     in 
the     directions    m  P,    m  P\ 
m  P" ,  not  in  the  same  plane ; 
the  resultant    will   be   repre- 
sented in  intensity  and  direc- 
tion    bv    the    diagonal    of    a 
parallelopipedon,  constructed 
upon   the    lines    representing 
the  directions  and  intensities 
of  these    components.      For, 
lay    off    the    distances    mA, 
m  C,  and    m  E,   proportional 
to  the  intensities  of  the  com- 
ponents   which   act   in    the   direction  of  these  lines,  and  construct  the 
parallelopipedon    E B  ;    the    resultant    of    the    components    P'  and    P 
will,    §82,    be   represented  by  the  diagonal  m  B,  of  the  parallelogram 
m  A  B  C ;    and  the  resultant  of  this  resultant  and  the  remaining  com- 
ponent P",  will  be  represented  by  the  diagonal  m  D  of  the  parallelo- 
gram   E  m  B  D,   which   is    that  of  the  parallelopipedon. 

§  86. — If  the  forces  act  at  right  angles  to  each  other,  the  parallel* 
opipedon  will  become  rectangular,  and  the  intensity  of  the  resultant, 
denoted   by  R,  will   become  known  from  the  formula 


53 


ELEMENTS    OF    ANALYTICAL     MECHANICS. 


R  =  yfp*  4-  W*  -f  P'n  ; 


and  if  the  angles  which  the 
direction  of  the  resultant 
makes  with  those  of  the 
forces  P,  F  and  P",  be 
represented  by  «,  b,  and  c, 
respectively,  then  will 

R  cos  a  =  P, 
R  cos  b  =  P\ 
R  cos  c  =  F\ 


Let  three  lines  be  drawn  through   the  point  of  application    m',   of 
the  force  P\  parallel  to  any  three  rectangular  axes  x,  y, z\  and  denote 
*  y  a'.  p\  y' .  the  angles  which 
ih«   direction    of     this    force 
makes    with    these    axes    res- 
pectively ;   then  will 


P'  cos  af, 
P'  cos  £', 
P'  cos  y\ 


Z 


A 


X 


r/ 


be  the  components  of  the  force  P\  in  the  direction  of  the  axes,  and 
they  will  act  along  the  lines  drawn  through  the  point  m\  These  are 
the  same  as  the  terms  composing  in  part  Equations  (J5),  and  as  the 
efl'cct  of  the  components  is  identical  with  that  of  the  resultant,  these 
components    may   always    be   substituted  for  the  force  P\      The  same 

for    the     forces    of    inertia,  and     m* — ->    m — -?   and  m»— -■>    denote    the 

dt*        dt2  at1 

components    of    this    force    \n    the    directions    ->f   the    axes. 


MECHANICS     OF     SOLIDS. 


59 


§87. — Examples. — 1.  Let  the  point  wi,  be  solicited  by  two  forces 
whose  intensities  are  9  and  5,  and  whose  directions 
make  an  angle  with  each  other  of  57°  30'.  Re- 
quired the  intensity  of  the  force  by  which  the 
point  is  urged,  and  the  direction  in  which  it  is 
compelled  to  move. 

Fir->t,  the  intensity  ;    make  in  Equation  (56), 

P   =  9, 
P"  =  5, 
S  =  57°  30' ; 

and  there  will  result, 


R  =  V  81  +  25  +  90  x  0,  537  =  12,422. 

Again,  substituting    the    values  of  j,  P'  P"  and  R  in  the  first  of 
Equations  (59),  we  have, 

,       5  X  sin  57°  30' 


sin  <p 


12,422 


or. 


<p'  =  19°  50'  35"  nearly, 


which  is  the  angle  made  by  the  direction  of  the  force  9  with  that  of 
the  resultant. 

2. — Required  the  angle  under  which  two  equal  components  should 
act,  in  order  that  their  resultant  shall  be  the  ntk  part  of  either  of  them 
separately. 

Bv  condition,  we  have 


hence, 


P'  =  P"  =  nR; 


p>  +  p"  +  R  nR  +  nR  +  R        (2n  -f  1)  H 

2         -  -  *  -  2  -         2        ; 


and,  Equation  (58), 


.     .               /(S  -  P')  (S  -  P") 
mh*  =  V ]Fp* ' 


60  ELEMENTS    OF     ANALYTICAL     MECHANICS. 

which  reduces  to 

sin**   =   ±JL 

If  n   be  equal   to   unity    or  the   resultant  be  equal  to  either  force, 


<p  =  60°, 
and,  §83,   the  angle  of  the  components  should   be  120°. 

3. — Required  to  resolve  the  force  18  =  «,  into  two  components 
whose  difference  shall  be  5  as  b,  and  whose  directions  make  with 
each  other  an  angle  of  38°  as  8.  Also,  to  find  the  angle  which  the 
direction   of  each  component  makes  with  that  of  the  resultant. 

Writing  a  for  R   in   Equation   (56),   we  have, 

P'2  +  P>n  +  o  p'  p"  cos  s  =  a\ 

and  by  condition, 

P'  -  p»  =  b     ,     .     .     ,     .     .     (c). 

Squaring  the  second  and  subtracting  it  from  the  first,   we  get 

27>'P"  (1  -f  cos  S)  as  a2  -  62  ; 

which,   replacing  (1  -f  cos  £)   by  2  cos2  ^  5,  reduces  to 

«2  -  62 


4P'  P"  ^ 


cos2  ^  8 


This  added  to  the  square  of  the  Equation   ( c ),  gives 


V  cos-*  ^  o 

from  which  and  Equation  (c)  we  finally   obtain, 


V 

-   /,2  (I 
COS2 

i 

COS2 

**) 

8 

V 

-  b2  (1 

— 

cos2 

**) 

which  are  the  required   components. 

To  find  the  angles  which  their  directions  make  with  the   resultant, 
we  have  from   Equations  (59), 

<p"  =  24°  ='  the  alible  wh'ch  P"  makes   with  the  resultant 


MECHANICS    OF    SOLIDS. 


61 


*iid, 


cpf  =  14°  =  angle  which  P'  makes  with  the  resultant. 


4. — Required  the  angle  under  which  two  components  whose    inten- 
sities are  denoted  by  5   and  7  should   act,  to  give  a  resultant  whose 

intensity  is  represented  by  9. 

Am.  84°    15'    39" 


5. — From  Equation  (56)  it  appears  that  the  resultant  of  two 
components  applied  to  the  same  point,  is  greatest  when  the  angle 
made  by  their  directions  is  0a,  and  least  when  180°.  Required  the 
.ingle  under  which  the  components  should  act,  in  order  that  the 
resultant  may  be  a  mean  proportional  between  these  values;  and 
also  the  angle  which  the  resultant  makes  with  the  greater  component. 
Call  P',  the  greater  component. 

-i       P" 

Am.  6  =  cos    —  —  • 


V 


sin* 


-i  T" 


6. — Given  a  force  whose  intensity  is  denoted  by   17.     Required   the 
two  components   which  make  with  it  angles  of  27°  and  43°. 

§  88. — The  theorem  of  the  parallelogram  of  forces,  just  explained, 
i-nables  us  to  determine  by  an  easy  graphical  construction  the  in- 
tensity and  direction  of  the  resultant  of  several  forees  applied  to  the 
same   point. 

Let  P\  P",  f»\  &c,  be 
several  forces  applied  to  the  jr 

same  point  m.  Upon  the 
directions  of  the  forces,  lay 
off  from  the  point  of  ap- 
plication distances  propor- 
tional to  the  intensities  of 
the  forces,  and  let  these  dis- 
tances represent  the  forces. 
From  the  extremity  Pr  of 
the   line    m  P\    which    repre- 


/>" 


62  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

sents  the  first  force,  draw  the  line  P'  n  equal  and  parallel  to  m  PA 
which  represents  the  second,  then  will  the  line  joining  the  extremity 
of  this  line  and  the  point  of  application,  represent  the  resultant  of 
these  two  forces.  From  the  extremity  n.  draw  the  line  n  n'  equal 
and  parallel  to  m  P'"  which  represents  the  third  force ;  m  n'  will 
represent  the  resultant  of  the  first  three  forces.  The  construction 
being  thus  continued  till  a  line  be  drawn  equal  and  parallel  to 
every  line  representing  a  force  of  the  system,  the  resultant  of  the 
whole  will  be  represented  by  the  line,  (in  this  instance  m  n"),  join- 
ing the  point  of  application  with  the  last  extremity  of  the  last 
line  drawn.  Should  the  line  which  is  drawn  equal  and  parallel  to 
that  which  represents  the  last  force,  terminate  in  the  point  of  appli- 
cation, the   resultant  will    be   equal    to   zero. 

The  reason  for  this  construction  is  too  obvious  to  need  expla- 
nation. 

§89. — If  the  forces  still  be  supposed  to  act  in  the  same  plane, 
but  upon  different  points  of  the  plane,  the  first  of  Equations  (49) 
takes   the   form, 

Yx  -  Xy  =  2  IP'  (cos  (3'  x'  -  cos  a'  y')  ], 

thus,  differing  from  Equation  (55),  in  giving  the  equation  of  the  line 
of  direction  of  the  resultant  an  independent  term,  and  showing 
that  this  line  no  longer  passes  through  the  origin.  It  may  be  con- 
structed from    the  above  equation. 

§  90. — To  find  the  resultant  in  this  case,  by  a  graphical  construc- 
tion, let  the  forces  P\ 
P",   P>"    &c,    be    ap-       j,  ^ 

plied  to  the  points  m\       \  ^r ~P  *" 

m",   m'" ,  &c,   respec-  ^»  /   \    / 

tively.       Produce    the  /    \  _./- {,„ 

J  &  /  \  Oj  K 

directions  of  the  forces  -      ^r  y©   V  /\ 

Pf   and    P"   till    they  \/  \    /       \ 

meet   at    0,   and   take  /  J?  \ 

this    as   their  common      |2>"  \ 

point    of    application  ; 

lay  off   from    0,    on   the     ines   of  direction,  distances   0  S  and   0  ST. 


MECHANICS     OF     SOLIDS.  63 

proportional  to  the  intensities  of  the  forces  P'  and  P",  and  construct 
the  parallelogram  0  S  R  S',  then  will  0  R  represent  the  resultant  of 
these  forces.  The  direction  of  this  resultant  being  produced  till  it 
meet  the  direction  of  the  force  P"\  produced,  a  similar  construction 
will  give  the  resultant  of  the  first  resultant  and  the  force  P'", 
which  will  be  the  resultant  of  the  three  forces  P\  P"  and  P'" ; 
and   the   same  for    the    other  forces. 

OF   PARALLEL   FORCES. 

§91. — If  the  forces  act  in   parallel   directions, 

cos  a  =  cos  a"  =  cos  a!"  =  <fcc, 
cos  /3'  =  cos/3"  =  cos  j3'"  =  &c, 
cos  y'  =  cos  y"  =  cos  y  "'  =   <fcc, 

and  Equations  (41)  become, 

X  =  (Pf  +  P"  +  P"'  +  &c.)  cos  a', 
Y  =  (Pr  +  P"  +  P'"  +  &c.)  cos  /3f, 
Z  =s.  (P'  +  P"  +  P"r  +  &c.)  cos  y' ; 

these  values  in  Equation  (47)  give, 

R  =  ±       -/(P7  +  P'7  +  -P"'  +  &c.)2  (cos2  a'  +  cos2  /3'  +  cos2  /), 

but, 

cos2  a'  -f  cos2  /3'  -f  cos2  y'  =  1 ; 
hence, 

R  =  P'  +  P"  +  P'"  +  &c.  •     •     •     •     •     (60) 

If  some  of  the  forces  as  P",  P"\  act  in  directions  opposite  to 
the  others,  the  cosines  of  a"  and  a'"  will  be  negative  while  they 
have    the    same   numerical   value ;  and   the   last  equation  will  become 

R  =  P'  -  P"  -  P'"  4-  &c. 

Whence  we    conclude,  that   the  resultant   of  a   number   of  parallel 
forces    is    equal   in    intensity    to    the    excess   of    the   sum    of   the    inten 
sities    of    those     which    act     in     one    direction     over     the     sum    of    tht 
intensities    of  those   which    a~t    in    the   opposite   direction. 


64  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

§92. — The  values  of  P,  X,   Y  and    Z  being    substituted    in  Equa 

tions  (48)  give, 

IP'  4  P"  4  P"  4  &c.)  cos  a' 
cosa  =  -F,  +  p„  +  pf„  +  &c =  cosa, 

(P'  +  P"  4  P'"  +  &c)  cos  /3' 
cos  6  =  -Htt— — r-7— — „,„    ,  ' =  cos  (3\ 


"  /* 

+  P" 

4-  P'" 

4-  &c. 

(i» 

4-  P" 

4  P'" 

4  &c.) 

cos 

V 

"  P' 

4-  P" 

4  P'" 

4-  &c. 

.  (^ 

4  P" 

4-  P"' 

4-  &c.) 

cos 

7' 

008  *  -   >  +  P"  +  P'"  +  &c.' "  -  C°S  y'- 

The  denominator  of  these  expressions,  being  the   resultant,  is  essen- 
tially positive ;    the   signs   of  the    cosines    of  the   angles    a,  b   and   c. 
will,  therefore,  depend   upon   the   numerators ;    these   are   sums   of  the 
components  parallel  to  the  three  axes. 

Hence,  the  resultant  acts  in  the  direction  of  those  forces  lohost 
cosine  coefficients  are  negative  or  positive  according  as  the  sum  of  the 
former  or  latter  forces  is  the  greater. 

§  93. — Equations  (49),  which  are  those  of  the  resultant,  become, 
itfter  replacing  Xy  Y,  and  Z,  by  their  values  in  Equations  (41), 

By .  cos  a  —  Rx .  co.^  b  4  cos  /3' .  2  P V  —  cos  a' .  2  P'y'  =  0? 

Bx .  cos  c  —  Bz  .  cos  a  4-  cos  a' .  2  P'z'  —  cos  y' .  2  P'x'z=  0, 

Pz.cos  b  —  By  .cose  +  cos  y' .  2  P'y'  -  cos  £'.2  PV=  0  j 
aihl  because, 

cos  a  =  cos  a', 

cos  b  =  cos  /3', 

cos  c  =  cos  y  ; 

vre  hare, 

(By  -  IPi/)  .  cos  a'-  (Bx  -  2?V) .  cos  fPss  0, 
(ite  -  2PV).cos  y-  (Bz  -  IP'z')  .  cos  a.'  =.  0 
(Bz  -  2PY) .  cos  #'-  (By  -  2 P';f) .  cos  y'  =  0 ; 
raid  because  a',  ,8'    and  7',  are  connected  only  by  the  relatioi 


MECHANICS    OF    SOLIDS 


65 


cosV  +  cos"/3"'  -}-  cos'/  =  1  ; 

either  two  of  the  cosines  of  these  angles  are  wholly  arbitrary,  and 
from  the  principle  of  indeterminate  co-efficients,  we  have,  by  dispens- 
ing with  the  sign  £  and  writing  out  the  terms, 


Rx  =  P'x'  +  P"x"  -h  P"'x'n  +  &c  ' 
Ry  =  P'y'  +  P"y"  +  P"'y'"  -f  &c.     K  • 
i?2    =   PV   +    P"*"   +   P'"z'"   -f   &C.   , 


(61) 


The.  forces  being  given,  the  value  of  R,  §91,  becomes  known, 
and  the  co-ordinates  x,  y,  z,  are  determined  from  the  above  equations  ; 
these  co-ordinates  will  obviously  remain  the  same  whatever  direction 
be  given  to  the  forces,  provided,  they  remain  parallel  and  retain  the 
same  intensity  and  points  of  application,  these  latter  elements  being 
the  only  ones  upon  which  the  values  of  x,  y,  z,  depend. 

The  point  whose  co-ordinates  are  x,  y,  z,  which  is  a  point  of 
application  of  the  resultant,  is  called  the  centre  of  parallel  forces,  and 
may  be  defined  to  be,  that  point  in  a  system  of  parallel  forces  through 
which  the  resultant  of  the  system  will  always  pass,  whatever  be  the 
direction  of  the  forces,  provided,  their  intensities  and  points  of  appli- 
cation remain    the  same. 


§  94. — Dividing  each  of  the  above  Equations  by  R,  we  shall  have 

p'x'  +  py  +  p'"x'"  +  &c 
p>  ~+_p',  +  p'  ••  •  - 


X   = 


y  = 


z  = 


""  +  &c. 
py  +  P"y"  4-  P'"y"'  +  &c- 

P'z'  -f  P"z"  -f  P'"z'"  +  &c 
P'  4-  P"  -f  P'"  4-  &c. 


(6-2) 


Hence,  e/7Aer  co-ordinate  of  the  centre  of  a  system  of  parallel  force* 
is  equal  t'  the  algebraic  sum  of  the  products  which-  result  from  multi- 
plying the  intensity  of  each  force  by  the  corresponding  co-ordinate  of  its 
point  of  application,  divided  by  the  algebraic  sum  of  the  forces. 

If  >Le   points  of  application  of  the    forces   be   in    the    same    plane. 


66 


ELEMENTS    OF     ANALYTICAL    MECHANIC©. 


the   co-ordinate   plane   xy,   may   be   taken   parallel    to   this   plane,   in 
winch  case 


and, 


z  = 


z'  =  z"  =  «"'  =  z""  &c. ; 


(P'  +  P"  +  P'"  -f  &c.)  z' 
p,  +  p„  +  p,„  +  &c> 


=  •: 


from  which  it  follows  that  the  centre  of  parallel  forces  is  also  in  this 
plane. 

If  the  points  of  application   be   upon   the   same   straight  line,  take 
the  axis  of  x  parallel  to  this  line ;  then  in  addition  to  the  above  results 


we  have 


y'  =  y" 


'j 


ttt 


&c. ; 


and, 


(Pf  +  P"  +  P'"  +  &c)  y' 
V  -      P'  +  P"  +  P'"  +  &c.     ~  V  ' 


whence,  the  centre  of  parallel  forces  is  also  upon  this  line. 

§  95. — If  we  suppose  the  parallel  forces  to  be  reduced  to  two,  viz. 
P*  and  P",  we  may  assume  the  axis  x  to  pass  through  their  points 
of  application,  and  the  plane  xy  to  contain  their  directions,  in  which 
case,  Equations  (60)  and  (61)  become, 

R  =  P'  +  P" 
Rx  =  P'x'  +  P"x" 
z  =  0   and   y  =  0. 

Multiplying  the  first  by  x\  and  subtracting 
the  product  from  the  second,  we  obtain 

R(x  -  x')  =  P"  (x"  -  O  .  .  (a) 

Multiplying  the  first  by  x"  and  sub- 
tracting the  second  from  the  product, 
we  get 

R  (x"  -x)  =  P'  (x"  -  x')     .     .     .     .     (  b  ) 

Denoting  oy  S'  and  S",  the  distances  from  the  points  of  application 


MECHANICS    OF    SOLIDS. 


67 


of  P'  and  P"  to  that  of  the  resultant,  which  are  x  —  x'  and  x 
respectively,  we  have 


x 


tt 


x'  =  Sf  +  S1 


ti 


and  from  Equations  (a)  and  (b),  there  will  result 

P'  :  P"  :  R  : :  S"  :  S'  :  S"  +  & 


(63) 


If  the  forces  act  in  opposite  directions,  then,  on  the  supposition 
that  P'  is  the  greater,  will 

R  =  P'  -  P" 
Rx  =  P'x'  -  P"x" 
z  =  0,   y  =  0. 

and  by    a    process    plainly   indicated    by 
what  precedes, 

P'  :  P"  :  R  : :  flf  :  5"  :  S"  -  S'.  .     (64). 

From  this  and  Proportion  (63),  it  is 
obvious  that  the  point  of  application  of 
the  resultant  is  always  nearer  that  of  the 
greater  component;    and    that    when    the 

components  act  in  the  same  direction,  the  distance  between  the  point 
of  application  of  the  smaller  component  and  that  of  the  resultant,  is 
less  than  the  distance  between  the  points  of  application  of  the  com- 
ponents, while  the  reverse  is  the  case  when  the  components  act  in 
opposite  directions.  In  the  first  case,  then,  the  resultant  is  between 
the  components,  and  in  the  second,  the  larger  component  is  always 
between  the  smaller  component  and  the  resultant. 

And  we  conclude,  generally,  that  the  resultant  of  two  forces  which 
solicit  two  points  of  a  right  line  in  parallel  directions ;  is  equal  in  inten- 
sity to  the  sum  or  difference  of  the  intensities  of  the  components,  accord- 
ing as  they  act  in  the  same  or  opposite  directions,  that  it  always  acta 
in  the  direction  of  the  greater  component,  that  its  line  of  direction  is 
contained  in  the  plane  of  the  components,  and  that  the  intensity  of  either 
component  is  to  that  of  the  resultant,  as  the  distance  between  the  point 
of  application  of  the  other  component  and  that  of  the  resultant,  is  to 
the  distance  between  the  points  of  application   of  the  components. 


68 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


** 


TmT 


§"96. — Examples. — 1.  The  length  of  the  line  mf  m"  joining  the 
points  of  application  of  two  parallel  forces 
acting  in  the  same  direction,  is  30  feet ;  the 
forces  are  represented  by  the  numbers  15 
and  5.  Required  the  intensity  of  the  re- 
sultant, and  its  point  of  application. 

R  =  P'  4  P"  =  15  +  &  =  20  ; 
R    :  P'  : :  m"  m'  :  m"  o, 

20  :  15  ::  30  :  m"  o  —  22,5  feet, 

A  single  force,  therefore,  whose  intensity  is  represented  by  20,  applied 
at  a  distance  from  the  point  of  application  of  the  smaller  force  equal 
to  22,5  feet,  will  produce  the  same  effect  as  the  given  forces  applied 
•at  m"  and  mf. 

2. — Required  the  intensity  and  point 
•of  application  of  the  resultant  of  two 
parallel  forces,  whose  intensities  are  de- 
noted by  the  numbers  11  and  3,  and 
which  solicit  the  extremities  of  a  right 
line  whose  length  is  16  feet  in  opposite 
.directions. 


mf 


R  =  P'  -  P"  =  11        3  =  8, 
P*  -  P"  :  P'  : 


P'  .  m"  mf 


m"  m'  :  m"  o  =  ~'  "'    "'     =  22  feet. 


P'  -  P" 

3. — Given  the  length  of  a  line  whose  extremities  are  solicited  in 
the  same  direction  by  two  forces,  the  intensities  of  which  differ  by 
the  na  part  of  that  of  the  smaller.  Required  the  distance  of  the 
point  of  application  of  the  resultant  from  the  middle  of  the  line 
Let  2  /,  denote  the  length  of  the  line.     Then,  by  the  conditions, 

F  =  P"  4  -P"  =  (!^)  P" 

V      ft     /  n 

'2/i  4  1 


(*l±A)  p" 


P"  ::2l:m'o  = 


2nl 


2n  4  1 


SO   z=z   I 


2nl 


1 


2n  4   1       2n  4  1 


I 


1 


MECHANICS     OF    SOLIDS 


69 


1 97. — The  rule  at  the  close  of  §05,  enables  us  to  determine  by  a 
very  easy  graphical  construction,  the  position  and  point  of  application 
of  the  resultant  of  a  number  of  parallel  forces,  whose  directions, 
intensities,  and  points  of  application  are  given. 

Let  P,  P',  P'',  P'",  and  Piv, 
be  several  forces  applied  to  the 
material  points  m,  m\  m",  m"\ 
and  miv,  in  parallel  directions. 
Join  the  points  m  and  m'  by  a 
straight  line,  and  divide  this  line 
at  the  point  o,  in  the  inverse 
ratio  of  the  intensities  of  the 
forces  P  and  P' ;  join  the  points 
o  and  m"  by  the  straight  line 
om",  and    divide    this   line  at  o', 

in  the  inverse  ratio  of  the  sum  of  the  first  two  forces  and  the  force 
P"  ;  and  continue  this  construction  till  the  last  point  miv  is  included, 
then  will  the  last  point  of  division  be  the  point  of  application  of  the 
resultant,  through  which  its  direction  may  be  drawn  parallel  to  that 
of  the  forces.  The  intensity  of  the  resultant  will  be  equal  to  the 
algebraic  sum  of  the  intensities  of  the  forces. 

The  position  of  the  point  o  will  result  from  the  proportion 


P  -{-  P'  :  Pf  :  :  m  m'  :  m  o  = 


P' .  m  m' 
P  +  P 


t  y 


that  of  o'  from 


P  +  P'  +  P"  :  P"  ::  om"  :  oo' 


P"  .om 


t     l       E>/'   » 


P+P*  +  P 


that  of  o"  from 


P  +  ^  +  P"-  P"f :  -  F"'  :  o' «'"  :  o'  o"  = 


P'"  .o'm"' 


P  +  P'+P"  -F 


n  > 


and  finally,  that  of  o'"  from 


P+P'-f  P"_J>"'  +  i>"  :  P»  ::o"m*:  o" o 


Ht 


PiT .  o"  m 


p  +  p>  +  p"-.p>"  +  l 


70        ELEMENTS    OF    ANALYTICAL    MECHANICS 


OF     COUPLES. 

§  98. — When  two  forces  P'  and  P"  act  in  opposite  directions,  the 
distance  of  the  point  o,  at  which  the  resultant 
is   applied,  from   the   point   m\  at  which  the  jjx 

component   P'  is   applied,  is   found  from  the 
formula 


m"m'.P" 

and   if  the   components   P'   and    P"   become 

equal,  the  distance   m' o   will   be   infinite,  and 

the  resultant,  zero.     In  other  words,  the  forces 

will   have  no  resultant,  and   their   joint  effect 

will  be  to  turn  the  line  m"m',  about  some  point  between  the  points 

of  application. 

The  forces  in  this  case  act  in  opposite  directions,  are  equal,  but 
not  immediately  opposed.  To  such  forces  the  term  couple  is  applied. 
The  perpendicular  distance  between  the  lines  of  direction  of  the  forces 
is  called  the  arm  of  the  couple  and  the  product  of  the  intensity  of 
either  force  into  the  arm  is  called  the  moment  of  the  couple. 

The  effect  of  a  couple  is  to  produce,  or  tend  to  produce,  a  motion 
of  rotation  about  a  line  perpendicular  to  the  plane  in  which  the  forces 
act,  which  line  is  called  the  axis  of  the  couple.  If  there  is  no  motion 
of  rotation,  the  direction  of  the  line  is  given  only,  but  if  motion 
takes  place,  both  the  position  and  direction  of  the  axis  are  determined. 

No  matter  where  we  assume  the  position  of  the  axis,  in  case  of 
no  motion,  the  rotatory  effect  will  always  be  measured  by  the  moment 
of  the  couple.  And  a  little  study  will  show  that  the  effect  of  a 
couple  on  a  rigid  body  will  not  be  altered,  whatever  be  the  position 
of  the  plane  of  the  couple,  provided  the  direction  of  the  axis  is  un- 
altered and  the  arm  and  forces  are  the  same,  and  in  general  it  may 
be  shown  that  a  couple  is  equivalent  to,  and  may  be  replaced  by, 
any  other  couple  whose  moment  is  equal  and  the  direction  of  its 
axis  is  the  same.  Now,  in  all  these  transformations,  the  arm  and  the 
forces  may  be  altered    in    position,  in  length,  and   in  magnitude,  and 


MECHANICS    OF    SOLIDS.  71 

the  plane  in  which  the  forces  act  may  occupy  any  one  of  its  parallel 
positions.  But  the  axis  and  the  moment  must  remain  the  same,  and 
these  latter  cannot  be  changed  without  altering  the  effect  of  the 
couple — the  former  has  a  fixed  direction  and  the  latter  is  a  fixed 
quantity. 

It  is  convenient  in  these  forces  of  rotation,  as  in  forces  of  trans- 
lation, to  have  geometrical  lengths  as  adequate  representatives ;  and 
such  we  shall  obtain  if  along  the  axis  we  take  lengths  containing  the 
same  number  of  linear  units  as  the  moment  of  the  couple  contains 
units  of  pressure.  Thus,  if  the  force  of  a  couple  is  4*  and  the  length 
of  the  arm  is  3,  the  moment  is  represented  by  12  ;  and  if  along  the 
axis  12  linear  units  be  measured,  this  length  is  a  full  and  adequate 
representation  of  the  couple ;  and  as,  moreover,  couples  may  be  right- 
handed  or  left-handed,  that  is,  have  positive  or  negative  signs,  so 
from  the  origin  of  the  axis  may  the  line  be  taken  in  one  or  the  other 
direction  and  thus  indicate  the  sign  of  the  couple.  Now  if  we  tech- 
nically call  this  line  the  moment  axis  of  the  couple,  in  contradistinc- 
tion to  the  direction  of  the  axis  called  the  rotation  axis,  it  will  indi- 
cate three  things,  viz.,  the  line  of  rotation,  a  finite  length,  measured 
from  a  given  point  on  the  line,  and  the  direction  in  which  it  is 
measured.  This  axis  then  fully  determines  all  the  circumstances  of 
the  couple. 

If  then  by  coaxal  couples  wre  understand  those  whose  rotation  axes 
are  in  same  direction,  and  by  coaxal  and  equimomental  couples  those 
that  are  statically  equivalent,  we  can  readily  demonstrate  the  following 
theorems,  viz. : 

1.  The  resultant  of  many  coaxal  couples  is  a  coaxal  couple  whose 
moment  is  equal  to  the  algebraic  sum  of  the  moments  of  the  com- 
ponent couples. 

2.  If  two  lines  meeting  at  a  point  represent  the  moment  axes  of 
two  couples,  that  diagonal  of  the  parallelogram  constructed  on  these 
lines,  which  passes  through  this  point  will  represent  the  moment  axis 
of  a  single  equivalent  couple. 

It  readily  follows  that  couples  may,  by  means  of  their  moment 
axes,  which  are  their  geometric  representatives,  be  resolved  and  com- 
pounded according  to  the  same  laws  as  forces  of  translation,  by  means 


72         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

of  their  equivalent  lines  of  action,  and  whatever  is  true  of  forces  of 
translation  is  also  true  of  forces  of  rotation,  as  exhibited  bv  the 
moment  axes  of  the  couples  which  are  their  geometric  representatives. 

§  99. — The  analytical  condition,  Equation  (46),  expressive  of  the 
existence  of  a  single  resultant  in  any  system  of  forces,  will  obviously 
be  fulfilled,  when 

X=0,      F=0,     and     Z  =  0. 

But  this  may  arise  from  the  parallel  groups  of  forces  whose  sums 
are  denoted  by  X,  Y,  and  Z,  reducing  each  to  a  couple.  These  three 
couples  may  easily  be  reduced  by  composition  to  a  single  couple, 
beyond  which  no  further  reduction  can  be  made. 


WORK     OF     THE     RESULTANT     AND     OF     ITS     COMPONENTS. 

§  100. — We  have  seen  that  when  the  resultant  of  several  forces 
is  introduced  as  an  additional  force  with  its  direction  reversed,  it  will 
hold  its  components  in  equilibrio.  Denoting  the  intensity  of  the 
resultant  by  R,  and  the  projection  of  its  virtual  velocity  by  dr,  we 
have  from  Equation  (27), 

—  Rdr  +  P  .  dp  +  P'  .  dp  +  P"  .  dp"  +  &c,  =  0, 
or, 

Rdr  =  P.dp  +  P'.  dp'  +  P"  .  dp"  +  &c,   .     .     .     (65) 

in  which  P,  P',  P",  &c,  are  the  components,  and  dp,  dp',  dp",  <fec, 
the  projections  of  their  virtual  velocities. 

§  101. — Now,  the  displacement  by  which  Equation  (2V)  was  de- 
duced was  entirely  arbitrary ;  it  may,  therefore,  be  made  to  conform 
in  all  respects  to  that  which  would  be  produced  by  the  components 
P,  P',  &c,  acting  without  the  opposition  of  the  force  equal  and 
contrary  to  their  resultant;  and  writing  dr  for  oV,  dp  for  dp,  <fcc, 
Equation  (65)  will  become 

Rdr  =  Pdp  +  P'dp  -f  P"dp"  +  &c.,    ....      (66) 


MECHANICS    OF    SOLIDS.  73 

and  integrating, 

fRdr  =  fPdp  +  fP'dp'  +  fP"dp"  +  &c,     .     .     (67) 

in  which  R,  P,  P' ,  <fcc,  may  be  constant  or  functions  of  r,  p,  p',  &e., 
respectively. 

From  Equations  (66)  and  (67)  it  appears  that  the  quantity  of 
work  of  the  resultant  of  several  forces  is  equal  to  the  algebraic  sum 
of  the  quantities  of  work  of  its  components. 

Again,  replacing  Pdp,  P'$p\  &c,  in  Equation  (65),  by  their  values 
in  Equation  (31),  and  writing  dr  for  oV,  dp  for  dp,  &c.y  we  find 

fRdr  =  flP.  cos  a .  dx  -f  fZP .  cos  0 .  dy  -f  flP  .  cos  y .  dz,   .     (68) 

in  which  R  may  be  constant  or  a  function  of  r;  P,  constant  or  a 
function  of  x,  y,  z,  &c. 

If  the  forces  be  in  equilibrio,  then  will  R  =  0,  and, 

IP .  cos  a .  dx  -f  2P .  cos  (3  .  c?y  -f  2P .  cos  y .  dz  =  0.  .     *     (69) 


WORK     OF     ROTATION. 

§  102. — It  is  now  apparent  that  in  the  transformation  of  Equa- 
tion (^4)  to  Equation  (40),  each  force  of  the  original  system  was 
replaced  by  its  three  components  in  directions  of  three  rectangular 
axes,  arbitrarily  assumed. 

The  components  parallel  to  either  axis  will,  §  43,  work  during  any 
motion  which  will  carry  their  points  of  application  in  the  direction 
of  that  axis,  and  will  cease  to  work  when  the  motion  becomes  per- 
pendicular to  the  same  line. 

Let  the  points  of  application  of  the  forces  move  in  lines  parallel 
to  the  axis  z ;  the  components  parallel  to  z  alone  can  work,  for  the 
paths  being  perpendicular  to  the  directions  of  the  other  components, 
the  work  of  the  latter  will  be  nothing,  because  the  projections  of  the 
paths  upon  their  lines  of  direction  will  be  zero.  The  elementary 
work  of  the  extraneous  forces  will,  in  this  case,  be  found  in  the  third 
term  of  Equation  (40),  and  equal  to 

(S  P  cosy)  .  6zt. 


74        ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Again,  let  the  points  of  application  turn  around  the  axis  z,  parallel 
to  the  plane  xy\  the  components  parallel  to  the  axes  x  and  y  alone 
can  work,  since  the  paths  will  be  perpendicular  to  the  components 
in  the  direction  of  z,  and  their  projections,  therefore,  zero.  The  ele- 
mentary work  in  this  case  will  be  found  in  the  fourth  term  of  Equa- 
tion (40),  and  equal  to 

[2  P  {x'  cos  (3  —  y'  cos  a]  6(f). 

Now  let  both  of  these  motions  take  place  simultaneously ;  that  is,  let 
the  points  of  application  move  in  the  direction  of  the  axis  z,  and  also 
turn  about  that  line  ;  all  the  components  will  work,  because  the  paths 
will  be  oblique  to  their  directions,  and,  therefore,  have  projections  of 
measurable  values.  The  amount  of  elementary  work  of  the  extraneous 
forces  will,  in  this  case,  be  found  in  the  third  and  fourth  terms  of 
Equation  (40),  and  equal  to 

[(2  P  cosy)]  .  <te,  +  [2  P  (x'  cos (3  —  y'  cosce)]  .  cty. 

The  same  remarks  apply  to  motion  in  the  direction  of  and  about 
each  of  the  other  axes. 

§  103. — The  rule  for  estimating  the  quantity  of  work  when  the 
motion  is  parallel  to  either  axis  or  to  a  right  line  oblique  to  the 
three  axes,  is  simple ;  that  for  getting  the  work  during  motion  about 
an  axis,  is  not  so  obvious.  Let  the  motion  take  place  around  the 
axis  z\  and  consider,  first,  the  work  of  the  force  P.  The  two  com- 
ponents of  this  force,  viz.,  P  cos  (3  and  P  cos  ee,  which  enter  the  fourth 
term  of  Equation  (40),  have  for  their  resultant  P  sin  y.  This  result- 
ant, §  81,  acts  in  a  plane  parallel  to  that  of  xy,  and,  therefore,  at 
right  angles  to  the  axis  z.  Denote  by  ec;  the  angle  which  this  result- 
ant makes  with  the  axis  ar;   then  will 

P  cos  a  =  P  sin  y  .  cos  a.  )  fi_   ■ 

.  .      '   t (70) 

P  cos  Q  =  P  sin  y  .  sin  at,  J 

and  these  values  in  the  term  P  (x'  cos/?  —  y'  cos«),  give 

P  (x'  cos  (3  —  y'  cos  a)  =  P  .  sin  y  (x'  sin  at  —  y'  cos  #,)  .     (71) 


MECHANICS    OF    SOLIDS. 


75 


From  the  point  of  ap- 
plication m  of  P,  draw 
the  line  m  A'  perpendicu- 
lar to  the  axis  z;  denote 
its  length  by  h\  and  its 
inclination  to  the  axis  x 
by  0'.  Multiply  and  di- 
vide Equation  (71)  by  h' 
and  reduce  by  the  rela- 
tions 


x'  y' 

j,  =  cos<p';     — ,  =  sin0'; 


then  will  result 


0 

P  {x'  cos  0  —  y'  cos  a)  =  P  sin  y  h'  (sin  a, .  cos  (p  —  cos  o/ .  sin  0)  =  P  sin  y  h'  sin  (a,—  00- 

Draw  from  J'  the  line  A'k'  perpendicular  to  the  direction  of  the  line 
Pm  (produced),  and  denote  its  length  by  k' ;    then  will 


h'  sin  (a,  —  0')  =  k', 


and  there  will  result 


P  (x'  cos  (3  —  y  cos  «)  =  P  sin  y  .  k' ; 


(72) 


the   same   also    occurs   for  the   forces  P\  P",  &c. ;    so   that  we   mav 
write,  omitting  the  accent  from  k, 

2  P  (x'  cos  fi  —  y  cos  a)  =  2  P  .siny  .  £;  .     .     .     (73) 

and  the  measure  of  the  elementary  work   due  to   rotation   about   the 
axis  z  will  be  given  by  either  member  of  the  Equation 

[2  P  (x'  cos  j3  —  y  cosee)]  d(f>  =  [2,  P  sin  y  .  k]  6(f>     .     (74) 


§  104. — So  that  in  estimating  the  work  due  to  rotation  alone 
about  the  axis  z,  each  force  is,  in  effect,  replaced  by  its  two  compo- 
nents, the  one  parallel,  the  other  perpendicular  to  that  line,  and  the 
former  is  neglected  because,  in  this  motion,  it  cannot  work. 


76         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

§  105. — The  product  obtained  by  multiplying  that  one  of  the  two 
components  of  a  force  which  is  perpendicular,  while  the  other  is 
parallel,  to  a  given  line,  into  the  perpendicular  distance  between  this 
line  and  that  of  the  force,  is  called  the  component  moment  of  the 
force  in  reference  to  the  line,  and  it  measures  the  capacity  of  the 
force  to  produce  rotation  about  that  line  as  an  axis. 

§  106. — The  line  in  reference  to  which  the  moment  is  taken,  is 
called,  in  general,  a  component  axis ;  the  perpendicular  distance  from 
the  axis  to  the  line  of  direction  of  the  force  is  called  the  lever  arm 
of  the  force ;  and  the  extremity  of  the  lever  arm  on  the  axis  is  called 
a  centre  of  the  moment. 

When  the  direction  of  the  force  is  perpendicular  to  the  axis,  the 
latter  is  called  the  moment  axis  of  the  force.  In  this  case  the  com- 
ponent  parallel  to  the  axis  becomes  zero,  and  the  normal  component 
the  force  itself. 

The  moment  of  the  resultant  of  several  component  forces,  taken 
in  reference  to  its  moment  axis,  is  called  the  resultant  moment.  The 
moments  of  the  component  forces  are  called  component  moments. 

§  107. — Changing  6<p  into  d<f>  in  Equation  (74),  we  may  write 

[2  P  (x'  cos  j3  —  y  cos  a)]  d(f>  =  [2  P  sin  y  .  k]  deb,    .     (74) 
or, 

As  P  (x'cosfi  —  y'  cos«)]  d(f>=   f[l  P.siny  .  k]  def)    .     (74)' 

Whence  it  appears,  that  the  elementary  quantity  of  work  a  force  will 
perform  during  the  motion  of  its  point  of  application  about  an  axis, 
is  equal  to  the  product  of  the  moment  of  the  force  into  the  differential 
of  the  path  described  at  the  unWs  distance  from  the  axis. 

g  108. — The  whole  quantity  of  work  will  result  from  the  integra- 
tion of  Equation  (74)'  between  limits.  In  this  integration  two  cases 
may  arise,  viz.,  either  the  moment  may  be  constant,  or  it  may  be 
variable.  In  the  first  case,  the  quantity  of  work  is  obtained  by  mul- 
tiplying the  constant  moment  into  the  path  described    by  a   point  at 


MECHANICS    OF    SOLIDS.  77 

the  unit's  distance  from  the  axis.  In  the  second,  the  force  may  bo 
constant  and  the  lever  arm  variable ;  the  force  variable  and  the  lever 
arm  constant ;  or  both  may  be  variable,  and  in  such  way  as  not  to 
make  their  product  constant.  In  all  such  cases,  relations  between  the 
intensity  of  the  force,  its  lever  arm,  and  the  path  described  at  the 
unit's  distance,  must  be  known  in  order  to  reduce,  by  elimination,, 
the  second  member  of  Equation  (*74)'  to  a  function  of  a  single 
variable. 

These  remarks  are  equally  true  of  the  forces  of  inertia.  The  in- 
tensities of  these  depend  upon  the  masses  of  the  material  elements 
and  their  degree  of  acceleration  or  retardation  ;  their  points  of  appli- 
cation are  on  the  elements  themselves;  the  elementary  arc  described 
at  the  unit's  distance  is  the  same  for  both  sets  of  moments,  and  the 
value  of  the  moment  of  inertia  depends  upon  the  distribution  of  the 
material  with  reference  to  the  axis  of  motion. 

The  moments  of  the  forces  which  urge  a  body  to  tarn  in  opposite 
directions  about  any  assumed  axis  must  have  contrary  signs. 

The  sign  of  P  siny  k',  or  its  equal  P  cos/3  .  x  —  P  cos«  .  y\  de- 
pends upon  the  angles  which  the  direction  of  the  force  makes  with 
the  axes,  and  upon  the  signs  and  relative  values  of  the  co-ordinates 
of  the  point  of  application. 

Let  the  angles  which  the  direction  of  any  force  makes  with  the 
co-ordinate  axes  be  estimated  from  the  positive  side  of  the  origin ; 
then,  if  the  angles  which  this  direction  makes  with  both  axes  be 
acute,  and  the  point  of  application  lie  in  the  first  angle,  P  cos/3  .  x' 
and  P  cos  a  .  y'  will  be  positive,  and  if  the  first  of  these  products 
exceeds  the  second,  the  moment  will  be  positive ;  but  if  the  latter  be 
the  greater,  the  moment  will  be  negative.  The  same  remarks  apply 
to  the  other  axes. 

Since  the  effect  of  the  moment  of  a  force  is  analogous  to  that  of 
a  couple,  and  since  the  measure  of  this  effect  depends  no  less  upon 
the  lever  arm  than  upon  the  intensity,  we  may,  as  in  couples,  repre- 
sent geometrically  the  value  of  the  moment  with  reference  to  any 
moment  axis,  by  taking  as  its  representative  a  length  on  the  axis,  in 
the  proper  direction,  equal  to  as  many  linear  units  as  there  are  units 
in  the  product  of  the  intensity  by  the  lever  arm. 


78         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

If  the  line  with  reference  to  which  the  moment  is  taken  is  a 
component  axis,  then  the  length  will  be  found  by  multiplying  the 
intensity  of  the  component  perpendicular  to  the  axis  by  the  lever  arm 
of  the  component. 


COMPOSITION      AND      RESOLUTION      OF      MOMENTS. 

§  109. — The  forces  being  supposed  to  act  in  any  directions  what- 
ever, join  the  point  of  application  of  the  resultant  R  and  the  origin 
by  a  right  line,  and  denote  its  length  by  H.  Multiply  and  divide 
each  of  the  Equations  (44)  by  H,  and  reduce  by  the  relations, 

x 

-jj  =  cos  £ 

-^  —  cos  £ 

z 
-Yj.  =  cos  e, 

XZ 

in  which  £,  £,  and  e,  denote  the  angles  which  the  line  H  makes  with 
the  axes  x,  y,  and  z,  respectively ;   then  will 

R  .  H .  (cos  b  .  cos  £  —  cos  a  .  cos  £)  =  L, 

R  .  H  .  (cos  a  .  cos  e  —  cos  c  .  cos  £)  =  M,     r    .     .     (75) 

R  .  H  .  (cos  c  .  cos  £  —  cos  b  .  cos  e)  =  N.    J 

Squaring  each  of  these  Equations  and  adding,  we  find 

f      cos2  b .  cos2  £  —  2  cos  b  .  cos  a .  cos  £.  cos  £  -f-  cos2« .  cos2  £ 

Rl .  H*  i  -f-  cos2  a .  cos2  e  —  2  cos  a .  cos  c .  cos  e .  cos  £  -f  cos2c  .  cos2  J 

<  -r-cos^c  .  cos2£ —  2  cos&.cosc  .cos^.cose  -f-  cos26.cos2e  > 

=  Z2  -f  if2  +  iV2 (76) 

But 

cos2  a  -+-  cos2  b  -f  cos2c  =  1, (77) 

cos2£-f  cos2£  +  cos2e  =  1,    .....     (78) 

cos  a  .  cos  £  +  cos  6  .  cos  £  -J-  cos  c  .  cos  e  =  cos  0,     .     (79) 


MECHANICS    OF    SOLIDS.  79 

the  angle  0  being  that  made  by  the  line  H  with  the  direction  of 
the  resultant. 

Collecting  the  coefficients  of  cos2  a,  cos2  6,  cos2c,  and  reducing  by 
the  following  relations,  deduced  from  Equation  (78) ;   viz.: 

cos2  e  -\-  cos3  £  =s  1  —  cos2  £ 
cos2  £  -f-  cos2  e  as  1  —  cos2  £, 
cos2 £  -f-  cos2  £  =  1  —  cos8  e, 
we  find, 

IP .  iT3 .  [1  —  (cos  a .  cos  £+  cos  6 .  cos  |-J~  cos  c ,  cos  e)2] =Z2  -f  3P  +  N* ; 

from  Equation   (79), 

1  —  (cos«»cos£  -}-  cos 6,  cos£  -^  cose,  cose)2  =  1  —  cos20  =  sin2</>; 

which  reduces  the  above  to 

i?2.  IP,  sin20  ss  Z2  +  JP  +  A'2. 

But  /T2.sin2^  is  the  square  of  the  perpendicular  drawn  from  the 
origin  to  the  direction  of  the  resultant;  it  is,  therefore,  the  square 
or  the  lever  arm  of  the  resultant  referred  to  the  origin  as  a  centre 
of  moments.  Denoting  this  lever  arm  by  R,  we  have,  after  taking 
the  square  root, 


R.X-  */£>  +  M*  +  N2 (80) 

That  is  to  say,  the  resultant  moment  of  any  system  of  forces  is  equal 
to  the  square  root  of  the  sum  of  the  squares  of  the  sums  of  the  com- 
ponent moments,  taken  in  reference  to  any  three  rectangular  axes  through 
the  point  assumed  as  the  centre  of  moments, 

§  110. — This  important  relation  is  evidently  the  same  as  that  of 
a  resultant  force  to  its  components,  and  it  is  clear  that,  if  we 
geometrically  represent  a  moment  by  the  diagonal  of  a  rectangular 
parallelopipedon,  then  will  its  sides  represent  the  component  moments. 
Equation  (80)  may,  therefore,  be  called  that  of  the  parallelopipedon 
of  moments. 


80         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

§  111. — Assuming  the  linear  representative  for  the  moment  of  a 
force  as  indicated  in  Article  108,  and  combining  the  results  that 
follow  with  Equation  (80)y  we  derive  in  succession  all  the  rules  for 
the  composition  and  resolution  of  moments,  and  they  are  perfectly 
analogous  to  the  rules  for  the  composition  and  resolution  of  forces. 
Thus,  representing  by  0„  0y,  and  Oz  the  angles  which  the  resultant 
axis  makes  with  any  three  rectangular  co-ordinate  axes  through  the 
centre  of  moments,   we  shall  have 

R.  K .  cos02=Z  1 

R.  K.  cos0y  =  Jf  \ (81) 

R  .K .  cos0,  =  iVj 

which  shows  that  the  component  moment  of  any  system  of  forces  in 
reference  to  any  oblique  axis  is  equal  to  the  product  of  the  resultant 
moment  of  the  system  into  the  cosine  of  the  angle  between  the  resultant 
and  component  axes. 

And  since  the  axis  z  may  have  an  infinite  number  of  positions 
»nd  still  satisfy  the  condition  of  making  equal  angles  with  the  result- 
ant axis,  we  see,  Equation  (81),  that  the  sum  of  the  moments  of  the 
forces  in  reference  to  all  component  axes  which  make  equal  angles  with 
thi  resultant  axis  will  be  constant. 

From  Equations  (81)  we  readily  obtain 

L  L 

cos  02  =  ^ — r-  =  — __  .     .     .     .     (82) 

M                     M 
cos0y  =  — — ^  =      . =3 ....     (83) 

N  N 

COS  0,  =r  — -,  =  -— ....      (84) 

R  .  K  V/,2  +  M2  +  j\T2 

whence  we  conclude  that,  the  cosine  of  the  angle  which  the  resultant 
axis  makes  ivith  any  assumed  line  is  equal  to  the  sum  of  the  moments 
of  the  forces  in  reference  to  this  line  taken  uj  a  component  axis  divided 
by  the  resultant  moment. 


MECHANICS    OF    SOLIDS.  81 

For  the  same  system  of  forces  and  the  same  centre  of  moments, 
it  is  obvious  that  R  and  K  will  be  constant ;  whence,  Equation  (80), 
the  sum  of  the  squares  of  the  sums  of  the  moments  in  reference  to  any 
three  rectangular  axes  through  the  centre  of  moments,  taken  as  com- 
ponent axes,  is  a  constant  quantity. 

§  112. — Denote  by  0rt  Qy,  0„  the  angles  which  any  component 
axis  makes  with  the  co-ordinate  axes  z,  y,  and  #,  respectively,  and  by 
6  the  angle  which  the  component  and  resultant  axes  make  with  each 
other,  then  will 

cos  6  =  cos  Oz .  cos  6Z  4-  cos  0y .  cos  6y  -j-  cos  Qx .  cos  Bz ; 

multiplying  both  members  by  R  .  K,  we  have 

R.K.cos  6  =  R.  iT.cos  Oz .  cos  6Z  -j-  R.K. cos  Oy .  cos  6y  -f  R.K.cos  &x .  co&B^ 

But,  Equations  (81), 

R .  K .  cos  Oz  =  Z, 
R  .  K .  cos  Qv  ss  Mi 

R.  K  .cosex=zW; 

which  substituted  above,  give 

R  .  K .  cos  5  ■=.  L  .  cos  dz  -f-  M  .  cos  Bv  -f  N .  cos  0X  „    „     (85) 

That  is  to  say,  the  component  moment  in  reference  to  any  assumed 
component  axis  is  equal  to  the  sum  of  the  products  arising  from  mul- 
tiplying the  sum  of  the  moments  in  reference  to  the  co-ordinate  axes  by 
the  cosines  of  the  angles  which  the  direction  of  the  component  axis 
makes  with  these  co-ordinate  axes,  respectively. 

TRANSLATION     OF     EQUATIONS     (B)     AND     ( C). 

§  113. — Equations  (B)  and  (C)  may  now  be  translated.  They 
express  the  conditions  of  equilibrium  of  a  system  of  forces  acting  in 
various  directions  and  upon  different  points  of  a  solid  body.  These 
conditions  arc  six  in  number;   viz. : 


*. 


82  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

! 

1. —  The  algebraic  sum  of  the  components  of  the  forces  in  each  of 
any    three   rectangular   directions   must  be  separately    equal   to   zero ; 

2.  —  The  algebraic  sum  of  the  moments  of  the  forces  taken  in  refer* 
ence  to  each  of  three  rectangular  axes  drawn  through  any  assumed 
centre  of  moments,  must  be  separately   equal   to  zero. 

If  the  extraneous  forces  be  in  equilibrio,  the  terms  which  measure 
the  forces  of  inertia  will  disappear,  and  these  conditions  of  equilibrium 
will   be  expressed   by 

2  P.  cos  «  =  0, 

2P   cos/3  =  0,   > (B) 

2  P.  cos  7  =  0;J 


2P.  (xf  cos  (3  —  y'  cos  a)  =  0, "* 
2  P.  (z\  cos  a  —  xf  cos  y)  ==  0, 
2  P .  (yf  cos  7  —  z   cos  j3)  =  0. 


in 


The  above  conditions,  which  relate  to  the  action  of  a  system  of 
forces  on  a  free  body,  are  qualified  by  conditions  of  constraint  that 
determine  the  possible  motion. 

§114. — If  the  body  contain  a  fixed  point ',  the  origin  of  the  mova- 
ble co-ordinates,  in  Equation  (40),  may  be  taken  at  this  point ;  in 
which  case  we  shall  have, 

$xt  =  0, 

*y,  f  o, 

H  =  0; 

and  it  will  only  be  necessary  that  the  forces  satisfy  Equations 
( C),  these  being  the  co-efficients  of  the  indeterminate  quantities  that 
do  not  reduce  to  zero.  Hence,  in  the  ease  of  a  fixed  point,  the 
sum  of  the  moments  of  the  forces,  taken  in  reference  to  each  of  three 
rectangular  axes,  passing  through  the  point,  must  separately  reduce  to 
zero. 

Should    the    system    contain    two  fixed  points,  one  of  the  axes,    as 


MECHANICS     OF     SOLIDS.  83 

that  of  x,  may   be  assumed    to    coincide    with  the  line  joining   these 
points,  in   which  case,  there  will  result  in   Equation  (40), 

6xt  ~  0,  i?  =0, 
8yt  =  0,  S±  =  0. 
Sz,  =  0, 

and  it  will  only  be  necessary  that  the  forces  satisfy  the  last  Equa- 
tion in  group  ( C) ;  or  that  the  sum  of  the  moments  of  the  forces  in 
reference  to  the  line  joining  the  fixed  points,   reduce  to  zero. 

If  the  system  be  free  to  slide  along  this  line,  8  xt  will  not  reduce 
to  zero,  and  it  will  be  necessary  that  its  co-efficient,  in  Equation 
(40),  reduce  to  zero  ;  or  that  the  algebraic  sum  of  the  components  of 
the  given  forces  parallel  to  the  line  joining  the  fixed  points,  also  reduce 
to  zero. 

If  three  points  of  the  system  be  constrained  to  remain  IE  a 
fixed  plane,  one  of  the  co-ordinate  planes,  as  that  of  xy,  may  be 
assumed  parallel  to  this  plane;   in  which  case, 

ty  a  0, 

8*  =  0, 
<*4,  =  0; 

and  the  forces  must  satisfy  the  first  and  second  of  Equations  (B). 
and  the  first  of  (C);  that  is,  the  algebraic  sum  of  the  components 
ef  the  given  forces  parallel  to  each  of  two  rectangular  axes  parallel  to 
the  given  plane,  must  separately  reduce  to  zeto,  and  the  sum  of  the 
moments  in  reference  to  an  axis  perpendicular  to  this  plane  must  reduce 
to  zero. 

CENTRE    OF    GRAVITY. 

§115. — Gravity  is  the  name  given  to  that  force  wnich  urges  all 
bodies  towards  the  centre  of  the  earth.  This  force  acts  upon  every 
particle  of  matter.  Every  body  may,  therefore,  be  regarded  as 
subjected  to  the  action  of  a  system  of  forces  whose  number  is  equal 
to  the  number  of  its  particles,  and  whose  points  of  application  have, 
with  respect  to  any  system  of  axes,  tie  same  co-ordina'es  as  thes* 
part  icles. 


84  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

The  weight  of  a  body  is  the  resultant  of  this  system,  or  the 
resultant  of  all  the  forces  of  gravity  which  act  upon  it.  and  is  equal. 
in  intensity,  but  directly  opposed  to  the  force  which  is  just  sufficient 
to  support  the  bod  v. 

The  direction  of  the  force  of  gravity  is  perpendicular  to  tin- 
earth's  surface.  The  earth  is  an  oblate  spheroid,  of  small  eccentri- 
city, whose  mean  radius  is  nearly  four  thousand  miles ;  hence,  as  tho 
directions  of  the  force  of  gravity  converge  towards  the  centre,  it  is 
obvious  that  these  directions,  when  they  appertain  to  particles  of 
the  same  body  of  ordinary  magnitude,  are  sensibly  parallel,  since 
the  linear  dimensions  of  such  bodies  may  be  neglected,  in  compari 
sOn  with  any  radius  of  curvature  of  the  earth. 

The  centre  of  such  a  system  of  forces  is  determined  by  Equa- 
tions (62),  §  94,    which  are 

_  P'x'  +  P"x"  +  P"V"  +  &c.  5     ] 
*'  ~  P'.rfc  P"  +]P'"  +  &e> 

_  FY  +  P"y"  +  P'"y'"  +  &c.        I  , 

Hi  -  pf   +  p»  +   p»>  +  &c  II 

_  P'z'  +  P"z"  -f-  P'"z'"  +  &c. 

*'  ~~       f +7"T^"'  +  &c.     ' 

in  which  xt  yt  zi%  are  the  co-ordinates  of  the  centre ;  P',  P'\  &c. 
the  forces  arising  from  the  action  of  the  force  of  gravity,  that  is, 
the  weights  of  the  elementary  masses  m\  m",  &c,  of  which  the 
co-ordinates  are  respectively  x'  y'  z\   x"  y"  z" ,  &c. 

This  centre  is  called  the  centre  of  gravity.  From  the  values  of 
its  co-ordinates,  Equations  (86),  it  is  apparent  that  the  position  of 
this  point  is  independent  of  the  direction  of  the  force  of  gravity  in 
teference  to  any  assumed  line  of  the  body;  and  the  centre  of  gravity 
of  a  body  may  be  defined  to  be  that  point  through  which  its  -weight 
always  passes  in  whatever  way  the  body  may  be  turned  in  regard  to 
the  direction  of  the  force  of  gravity. 

The  values  of  P\  P" ,  &c,  being  regarded  as  the  weights  w\  w'\ 
&c,  of  the  elementary  masses  m\  m'\  &c,  we  have,  Equation   (I), 

P'  =  w'  =  mY;    P"  r=  *"  -r.  m" g"  ;    P'"  =  w'"  s*  m'"  q'"  ,   &c. 


MECHANICS    OF    SOLIDS. 


85 


and,  Equations  (86), 


x.  = 


y» 


z.  = 


__  m'g'x'  +  m"g"x"  +  m"' g'"  x'n  -f  &c 


m'/  +  m"  g"  +  m'"/"  +  &c. 
ro^y'  +  m"  g"  y"  4-  mf"  g'"  y'"  +  &c. 

wt'/g'  +  m" g"  z"  +  m'"  g'"  J"  ±  &c 
»»y  +  »»f7p  +  m"'/"  4-  &c. 


....  {sr\ 


g  1 16. — It  will  be  shown  by  a  process  to  be  given  in  the  proper 
place,  that  the  intensity  of  the  force  of  gravity  varies  inversely  as 
the  square  of  the  distance  from  the  centre  of  the  earth.  The  distance 
from  the  surface  to  the  centre  of  the  earth  is  nearly  four  thousand 
miles  ;  a  change  of  half  a  mile  in  the  distance  at  the  surface  would 
therefore,  only  cause  a  change  of  one  four-thousandth  part  of  its 
entire  amount  in  the  force  of  gravity ;  and  hence,  within  the  limits 
of  bodies  whose  centres  of  gravity  it  may  be  desirable  in  practice  to 
determine,  the  change  would  be  inappreciable.  Assuming,  then,  the 
force  of  gravity  at  the  same  place  as  constant,  Equations  (87), 
become 

nV  4-  m"  x"  4-  »'"*"'  4-  &c   ^ 


xt  = 


yt  = 


m!  4-  m"  4-  i»"'  ■  ■  &c 
mr  y'  4~  m"  y"  4-  ni 


"'  y'"  4-  &c 


m 


t    i 


m 


a 


4-  m"'  4-  &c. 


zs  = 


m'  z'  4-  m"  z"  4-  m'"  z'"  +  &c. , 
m'  +  m"  4-  »*'"  4-  &c 


(88) 


from  which  it  appears,  that  when  the  action  of  the  force  of  gravity 
is  constant  throughout  any  collection  of  particles,  the  position  of  the 
centre  of  gravity  is  independent  of  the  intensity  of  the  force. 

§  117. — Substituting  the  value  of  the  masses,  given  in  Equation  (1)'. 
tnere  vail  result, 


*i  = 


yt  = 


*t  = 


v'd'x'  4-  v"d"  x"  4-  v"'  d 


iti  jnt  „tn 


4-  &c. 

—  1 


v'd'  4-  v"d"  4-  v"'d'"  4  &c. 

v'd'y'  +  v"d"y"  +  v"' d'"  y"'  4-  &c. 

v'  df  4-  v"  d"  4-  v'"  d'"  4-  &c.        ' 

v>d'g'  4-  v"  d"z"  4-  v'"  d'"  z'n  4-  &c 
v'  d'  4-  v"  d"  4-  v'"  d'"  4-  &c. 


(SQ) 


86 


ELEMENTS    OF    ANALYTICAL    MECHANICS 


and  if  the   elements  be  of  homogenous   density  throughout,  we   shalJ 
have, 

tt  =  d"  =  d"'  =  &c. ; 

Mid  Equations  (89)  become, 


*i  = 


y,  = 


*/  = 


v'x'  4-  v"x"  +.  v'"x'"  +  &c.   1 

— > 


v'  4-  v"  -t-  v'"  4-  &c. 

*V  4-  »*jg  4-  v"f  y"'  4-  &c. 

v'  4-  v"  4-  *>'"  4-  &c. 

pV  4-  gig  4-  p"'z'"  4-&c.. 
*'  4-  v"  4-  *'"  4-  &c.       ' 


'   > 


(90) 


wnence  it  follows,  that  in  all  homogeneous  bodies,  the  position  of 
the  centre  of  gravity  is  independent  of  the  density,  provided  the 
intensity  of  gravity  is  the  same  throughout. 

§  118. — Employing  the  character  2,  in  its  usual  signification,  Equa- 
tions (90),  may  be  written, 

2  (vx)    " 


*,= 


yt  = 


Z{vy) 


_  M>j 

and  if  the  system  be  so  united  as  to  be  continuous, 

,/pm  x.dV 


(91) 


** 

V 

5 

Jv"  y> 

dV 

y/ 

V 

5 

«> 

Jv"     Z  ' 

dV 

> 


m 


V 

§119. — If  the   collection   be   divided   symmetrically    by    the   plane 

ry,  then  will 

Z(vz)  =  0, 


MECHANICS    OF    SOLIDS. 


87 


and,  therefore, 


0; 


hence,  the  centre  of  gravity  will  lie  in  this  plane. 

If,  at  he  same  time,  the  collection  of  elements  be  symmetrically 
divided  by  the  plane  xz,  we  shall  have, 

2  (vy)  =  0, 

y,  =  ° ; 

the  collection  of  elements  will  be  symmetrically  disposed  about  the 
axis  x,  and  the  centre  of  gravity  will  be  on  that  line. 

Although  it  is  always  true,  that  the  centre  of  gravity  will  lie  in 
a  plane  or  line  that  divides  a  homogeneous  collection  of  particles 
symmetrically ;  yet,  the  converse,  it  is  obvious,  is  not  always  true, 
viz. :  that  the  collection  will  be  symmetrically  divided  by  a  plane  or 
line  that  may  contain  the  centre  of  gravity. 

Equations  (92)  are  employed  to  determine  the  centres  of  gravity 
of  all  geometrical  figures. 

THE   CENTRE   OF   GRAVITY    OF   LINES. 


§  120. — Let  s  represent  the  entire  length  of  an  arc  of  any  curve, 
whose  centre  of  gravity  is  to  be  found,  and  of  which  the  co-ordi- 
nates  of  the   extremities   are   x\  y' ',  z\  and  x",  y",  z". 

To  be  applicable  to  this  general  case  of  a  curve,  included  within 
the   given   limits,  Equations  (92)  become 


x,  = 


n*f    ,     i      in?     hf  " 

J       xdx.yj  !  +  _  +  — 


y,  = 


tt  — 


yda 

t 

c 

'V1  + 

dy2 
dx2 

dz2 
+  dx* 

z  d  x 

S 

c 

:.N/"l    + 

dy2 
dx2 

dz2 

+  dx* 
j 

(WSj 


88 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


ii»  which 


=  fl  d*s[ 


1  + 


dy2 
dx2 


+ 


dz2 
dli? 


(94) 


Example  1. — Find   the  position  of  the  centre   of  gravity   of  a   right 
line.     Let, 

y  —  a  x  +  ft  Z 

z  =  a! x  +  £', 

be  the   equations  of  the 
line. 

Differentiating,  substi- 
tuting in  Equations  (94) 
and  (93),  integrating  be- 
tween the  proper  limits, 
and  reducing,  there  will 
result, 


— X 


xt  = 


x'  4-  x 

2~ 


i£+£L  +  A 


zt  z= 


a',  (z'  +  x") 


2 


+  (3f, 


which  are  the  co-ordinates  of  the  middle  point  of  the  line  ;  x'  yr  z' 
and  x"  y"  z" ,  being  those  of  its  extremities ;  whence  we  conclude 
that  the  centre  of  gravity  of  a  straight  line  is  at  its  middle  point.   • 

Example  2. — Find  the  centre  of  gravity  of  the  perimeter  of  a  polygon. 

This  may  be  done,  according  to  Equations  (90),  by  taking  the  sum 
ol"  the  products  which  result  from  multiplying  the  length  of  each  side 
by  the  co-ordinate  of  its  middle  point,  and  dividing  this  sum  by  the 
length  of  the  perimeter  of  the  polygon.  Or  by  construction,  as  fol- 
lows . 

The  weights  of  the  several  sides  of  the  polygon  constitute  a  system 
of  parallel  forces,  whose  points  of  application  are  the  centres  of 
gravity  of  the  sides.  The  sides  being  of  homogeneous  density,  their 
weights  are  proportional  to  their  lengths.     Hence,  to  find    the   centn* 


MECHANICS    Ob     SOLIDS. 


89 


of  gravity  of  the  entire  polygon,  join  the  middle  points  of  any  two 
of  the  sides  by  a  right  line,  and  divide  this  line  in  the  inverse  ratio 
of  the  lengths  of  the  adjacent  sides,  the  point  of  division  will,  §  97, 
be  the  centre  of  gravity  of  these  two  sides ;  next,  join  this  point 
with  the  middle  of  a  third  side  by  a  straight  line,  and  divide  this 
line  in  the  inverse  ratio  of  the  sum  of  first  two  sides,  and  this  third 
side,  the  point  of  division  will  be  the  centre  of  gravity  of  the  three 
sides.  Continue  this  process  till  all  the  sides  be  taken,  and  the  lai^t 
point  of  division  will  be   the  centre  of  gravity  of  the   polygon. 

Find  the  position  of  the  centre  of  gravity  of  a  plane  curve. 

Assume  the  plane  of  xy  to  coincide  with  the  plane  of  the  curve, 
m  which  case, 

d  z 


d  x 


=  0, 


and  Equations  (93)  and  (94)  become, 


x.  ~ 


V,  = 


c 

xdx  W   1 

dy2 
dx2 

s 
ydx  y    1 

nX> 

L" 

,    dy>  ■ 
T  dx2 

8 

} 

r" 

d  x    k  /    1    - 

dy2 

(95) 


•         •         •         •        • 


(96) 


Example.  3~ — Find  the   centre  of  gravity  of  a  circular  arc. 

Take  the  origin  at  the  centre  of  curvature,  and  the  axis  of  y 
passing  through  the  middle  point  of  the  arc.  The  equation  of  the 
curve   is, 

y2  =  a2  —  z2, 

whence^ 

dr.  y 

which  substituted  in  Equations  (95), 


90 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


will  give  on  reduction, 


*,  =  o, 


y*  = 


a  (x'  +  *")  . 


s 


and   denoting   the  chord  of  the  arc  by  c  =  x'  +  x'\ 


x,  =  0, 


ac  . 


whence  we  conclude  that  the  centre  of  gravity  of  a  circular  arc  ts 
on  a  line  drawn  through  the  centre  of  curvature  and  its  middle  point, 
and  at  a  distance  from  the  centre  equal  to  a  fourth  proportional  to 
the   arc,  radius  and  chord. 

Example  4. — Find  the  centre  of  gravity  of  the  arc  of  a  cycloid. 

The   radius  of  the  generating  circle  being   a,  the  differential  equa 
tion  of  the  curve  is, 


dx  = 


y  -dy 


-y/2  ay  —  y2 


(a) 


the   origin  being  at  A,  and 

A  B  being  the  axis  of  x.  "^ 

Transfer  the  origin  to  C, 
and  denote  by  x\  y'  the  new 

co-ordinates,  the  former  being  estimated  in  the  direction  CD,  and  the 
latter  in  the  direction  DA.    Then  will 


and  therefore, 


y  =  2a  —  x\ 
x  ac  a  if  —  y' ; 


dx       dyf      ^         2  a  —  xf 


w 


MECHANICS    OF    SOLIDS.  91 

this,  in   Equations  (96)  and  (95),  gives,  omitting    the   accent   on    the 
variables, 

=  f» dx  Vn? 


*    = 


x,  =. 


Js  xdx\r\ 


s 


x 

y,  = 


f,"  ydx\l—x   * 


Integrating   the  first    two   equations   between   the   limits    indicated, 

and  substituting  the  value  of  s,  deduced  from  the  first,  in  the  second, 

we   have, 

*   =  2<v/2a(yV  -  -y/ x'), 

1     -/s"3  _  wx>z 

x   =  —  •  — : 

-       3   yV'   -  y*'  ' 

and  from   the  third  equation  we  have,  after    integrating  by  parts, 

sy,  =  2y/2a  (y  t/ x  —  f-y/xdy)\ 

substituting   the  value  of  dy,   obtained    from    Equation  (a)',    and   re- 
ducing, there  will  result, 


sVi  =2  V%  a  (y  V x  ~  fV%a  —  x.dx)9 
and  taking  the  integral  between  the  indicated  limits, 

^=2v/^[y(v/77-  V*)  +  §  (2a -*")*- f  (2  a  -*')*1: 
hence,  replacing  s  by  its  value,  and  dividing, 

Supposing  the  arc  *)  begin  at  (7,  we  have, 

*'  =  0, 
and, 

*,  =  **", 


92 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


If  the  entire  semi-arc  from  C  to  A  be  taken,  these  values  become, 

*,  =  |«i 

y,  =  a  (*  -  |). 

Taking  the  entire  arc  A  €  B,  the  curve  will  be  symmetrical  with  res 
pect  to  the  axis  of  x\  and  therefore, 

V,  =  0; 

hence,  the  centre  of  gravity  of  the  arc  of  the  cycloid,  generated  by  one 
entire  revolution  of  the  generating  circle,  is  on  the  line  which  divides 
the  curve  symmetrically,  a^d  at  a  distance  from  the  summit  of  the  curve 
equal  to  one-third    of  its   height. 


THE    CENTRE    OF     GRAVITY    OF     SURFACES. 

§  121. — Let  L  =  0,  be  the  equation  of  any  surface;  L  being  a 
function  of  x  y  z ;  then  will  dxdy,  be  the  projection  of  an  element 
of  this  surface,  whose  co-ordinates  are  xyz,  upon  the  plane  xy;  and 
if  d"  denote  the  angle  which  a  plane  tangent  to  the  surface  at  the 
same  point  makes  with  the  plane  xy,  the  value  of  the  element  itself 
will  be 


dx .  dy 

cos  6"  ' 

But  the  angle  which  a  plane 
makes  with  the  co-ordinate 
plane  xy,  is  equal  to  the 
angle  which  the  normal  to 
the  plane  makes  with  the 
axis  z,  and,,  therefore, 


cos  6"  -    ± 


JL 

dz 


v  <£>•  ♦  m  *  m 


1 

=   rfc  — 
W 


(97) 


MECHANICS     OF    SOLIDS. 


93 


and  hence,  in  Equations  (92),  omitting  the  double  sign, 

d  V  =  dx-dy  >  w,   .     .     .     .     .     .     .     (98) 

nni  those  Equations  become, 

»y'  /»*' 


*,  * 


i.   f 
Jy'Jx'w    x.dx.dy 


»y'  />*' 


y,  = 


Jy'L"     W   . 


y  dx .  dy 


.y'    /»*' 


s,  = 


f     f 

Jy" J  x"    W  .  Z  .dx. 


dy 


(99) 


in  which, 


,y>      nxt 


*=   V=    f ,,  f ,,  w.dx.dy  ,      ....     (100) 


w  being  a  function  of  ar,  y,  z. 

If  the  surface  be  plane,  the 
plane  of  xy  may  be  taken  in  the 
surface,  in  which  case, 

w  =  J. 

z  =  o, 

and  Equations  (99),  and  (100),  be- 
come, 


A     J" 


»y'  /»*' 


t/y    t/x 


dy  .xd x 


s 


.y'   /»*' 


y, 


Jy"Jx"  dx.ydy 


8 


(101) 


»*'  /»*' 


*  =  /„  [„  dx  .dy,      ......      (102) 

in    which    rho    integral    is    to    be    taken    first    with    respect   to  y,  and 


94 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


between  the  limits  y"  =  P  m"  and  y'  =  P  m' ;    then  in  respect  to  a\ 
between  the  limits  x"  =  A  P'\  and  x'  =  A  P\     lienor 


V 


xl  = 


J*"(y"  -y').xdx 


s 


Vi 


s 


_hL"(ym-yf2)dx 


=   fa"  ~y')d> 


(103) 


(104) 


y'  and  y",  denoting  running  co-ordinates,  which  may  be  either  roots 
of  the  same  equation,  resulting  from  the  same  value  of  .t,  or  they 
may  belong  to  two  distinct  functions  of  x,  the  value  of  x  being  the 
same  in  each.      For  instance,  if 

F   (xy)  =  0, 

be  the  equation  of  the  curve  n'  m"  n"  m\  it  is  obvious  that  between 
the  limits  x"  =  AP"  and  x'  =  A  P\  every  value  of  .r,  as  A  P. 
must  give  two  values  for  y,  viz.:  y"  =  Pm"  and  y'  as  Pm'.      Or  if 


F(xy)  =  0, 
F'(xy)  =  0, 

be  the  equations  of  two  distmct 
curves  m"  n"  and  m'  n\  referred 
to  the  same  origin  A,  then  will 
y"  and  y'  result  from  these 
functions  separately,  when  the 
same  value  is  given  to  x  in 
•each. 


A 


rt' 


X 


Example  1. — Required  the  position  of  the   centre   of  gravity    of  flu 
area  of  a  triangle. 


MECHANICS     OF     SO- LIDS. 


95 


Let  A  B  C,  be  the  triangle. 
Assume  the  origin  of  co-ordi- 
nates at  one  of  the  angles  A, 
and  draw  the  axis  y  parallel  to 
the  opposite  side  B  C.  Denote 
the  distance  A  P  by  x\  and 
suppose, 

y"  —  <**, 
y'  =  bx, 

to    be    the    equations    of  the    sides    A  C  and    A  B,    respectively,    then 
will 

y"  —  i'  =  («  —  h)  z, 


and, 


x,  = 


y>'2     _    y>2     _      (a2      _     £2)     ^ 


/    (a  —  b)  x2  dx 

Jxt  *       , 

~~3  *' 


y< 


J    (a  —  b)  x  dx 

if  (a*  -  V)x*dx  ■■  tfj 

V       (a  —  6)  #  rf  # 


whence  we  conclude,  ^a£  £/*e  centre  of  gravity  of  a  triangle  is  on  a 
line  drawn  from,  any  one  of  the  angles  to  the  middle  of  the  opposite 
side,  and  at  a  distance  from  this  angle  equal  to  two- thirds  of  the  line 
thus  drawn. 

Example  2. — Find  the  centre  of  gravity  of  the  area  of  any  polygon.- 

- 

From    any   one   of  the    angles  J 

as  A,  of  the  polygon,  draw  lines 
to    all    the    other     angles    except 

those  which  are  adjacent  on  either  \  W  x^ 

side;  the  polygon  will  thus  be 
divided  into  triangles.  Find  by 
the  rule  just  given,  the  centre  of 
gravity  of  each  of  the   triangles; 


96 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


join  any  two  of  these  centres  by  a  right  line,  and  divide  this  line  in 
the  inverse  ratio  of  the  areas  of  the  triangles  to  which  these  centres 
belong  ;  the  point  of  division  will  be  the  centre  of  gravity  of  these 
two  triangles.  Join,  by  a  straight  line,  this  centre  with  the  centre  of 
gravity  of  a  third  triangle,  and  divide  this  line  in  the  inverse  ratio 
of  the  sum  of  the  areas  of  the  first  two  triangles  and  of  the  third,  this 
point  of  division  will  be  the  centre  of  gravity  of  the  three  triangles. 
Continue  this  process  till  all  the  triangles  be  embraced  by  it,  and  the 
last  point  of  division  will  be  the  centre  of  gravity  of  the  polygon  , 
the  reasons  for  the  rule  being  the  same  as  those  given  for  the  deter- 
mination of  the  centre  of  gravity  of  the  perimeter  of  a  polygon,  it 
being  only  necessary  to  substitute  the  areas  of  the  triangles  for  the 
lengths  of  the  sides. 

Example  8. — Determine    the   position    of  the    centre  of  gravity  of  a 
circular  sector. 

The  centre  of  gravity  of  the  sec- 
tor will  be  on  the  radius  drawn  to 
the  middle  point  of  the  arc,  since  this 
radius  divides  the  sector  symmetri- 
cally. Conceive  the  sector  C  A  B,  to 
be  divided  into  an  indefinite  number 
uf  elementary  sectors :  each  one  of 
these  may  be  regarded  as  a  triangle 
whose  centre  of  gravity  is  at  a  dis- 
tance   from    the    centre    C,    equal    to 

two  thirds  of  the  radius.  if,  therefore,  from  this  centre  an  arc  be 
described  with  a  radius  equal  to  two-thirds  the  radius  of  the  sector, 
this  arc  will  be  the  locus  of  the  centres  of  gravity  of  all  the 
elementary  sectors ;  and  for  reasons  already  explained,  the  centre  of 
gravity  of  the  entire  sector  will  be  the  same  as  that  of  the  portion 
of  this  arc  which  is  included  between  the  extreme  radii  of  the  sector. 
Hence,  calling  r  the  radius  of  the  sector,  a  and  c  its  arc  and  chord 
respectively,  and  xt  the  distance  of  the  centre  >f  gravity  from  the 
centre   C,  we  have, 


x. 


lr-lc 


2      r .  c 


<t. 


3 


a 


MECHANICS     OF     SOLIDS. 


n 


The  centre  of  gravity  of  a  circular  sector  is  therefore  on  the  radius 
drawn  to  the  middle  point  of  the  arc  of  the  sector,  and  at  a  distance 
from  the  centre  of  curvature  equal  to  two-thirds  of  a  fourth  propor- 
tional   to    the   arc,  chord  and   radius  of  the   sector. 

Example  4. — Find  the  centre  of  gravity  of  a  circular  segment. 

Assume  the  origin  at  the  centre  C, 
and  take  the  axis  x  passing  through  the 
middle  point  of  the  arc,  the  centre  of 
gravity  in  question  will  be  on  this  axis, 
and,    therefore, 

y,  =  o. 

Let  A  B  HA  be  the  segment,  and 


y  =z  ±  yf  a1  —  x2, 

the  equation  of  the  circle,  the  origin  being 
at    the  centre   C,  then  will 


y''  =         yfaJ~  ■  x\ 


—  \/a- 


'  ■» 


and,  Equations  (103)  and  (104), 

2Ja    Va2  ~  ^.x.dx  }{a'< 


x'rf 


_  %f*   ^,/2_  x*.dx    =      a2  (^  -  sin   '     X-)   -  xf  yftfi 


X 


"2 


9    being   the    area  of   the    entire    segment.      Denoting   the   chord  AB 
by  c,  we  have, 


yV  -*"=£<;; 


whence. 


x.  = 


12.5 


and    we    conclude,    that    tlie    centre    of  gravity    of    a    circular    segment, 
its    on    the    radius    draion    to    the    middle    of  the  arc,  and  at  a   distance 
from    the    centre     equal     to    the    cube    of   the     chord,    divided    by    tweht 
'imcs    the   area    of  the   segment. 


98 


ELEMENTS     OF     ANALYTICAL    MECHANICS 


Replacing   the    value   of  s,  and   supposing   x'    tc   be   zero,  in   which 
case    the   segment  becomes   a   semicircle,  we   shall  find, 


Va, 


4a 
4        3* 


§  122. — If  the  surface   be   one  of  revolution,  about   the   axis  x  for 
instance,  it  will  be   symmetrical  with  respect  to  this  axis ;  hence, 

y,  =0;     ffsO; 

and   if  F(xy)  =  0,    be   the   equation    of    a   meridian    section   in    the 
plane    xy,    then  will  the   area  of  an    elementary    zone   comprised  b» 
tween   two   planes   perpendicular   to    the    axis  of  revolution    be, 

2< . y .  x'dx2  -f-  d y2, 
and  therefore,  Equations    (92), 


xt  =  2* 


rx'  IT       dy2      , 


=  2./::y.v/i4-g.^ 


•    •    « 


(105) 
(106) 


Example  1. — Find 
the  position  of  the 
centre  of  gravity  of 
a  right  conical  sur- 
face. 

The  equation  of 
the  element  in  the 
plane  xy,  is,  assum- 
ing the  origin  at  the 
vertex, 


hence, 


x.  = 


y  =  ax; 

2  *  f  °„  a x2  d x  y/\  -f  a2 
2  if   I  tt  ax   dx  -y/  1  -+-  a2 


SS    X 


tt 


MECHANICS    OF    SOLIDS. 


99 


Example  2. — Required  the  posi- 
tion of  the  centre  of  gravity  of 
a   spherical  zone. 

Assuming  the  origin  at  the 
centre,  the  equation  of  the  me- 
ridian   curve   is, 

.2. 


y2  =  a2 


x* 


whence, 


y  d  y  =   —  x  d  x, 


dy2 
d& 


y1' 


and, 


x. 


r; 


ax  d x 


x 


"'l    x'2 


S, 


ad  x 


2  (*"  -  x') 


x"  +  xf 

—   i  ■  -  ■■  • 

2 


Hence,  the  centre  of  gravity  of  a  spherical  zone,  is  at  the  middle 
point  of  a  line  joining  the  centres  of  its  circular  bases.  And  in  tlif 
case  of  one    base    it  is   only^  necessary  to   make   x"  =  o,  which  gives, 


*,  = 


x'  -{-  a 


So  that  the  centre  of  gravity  of  a  zone  of  one  base  is  at  the  middle 
of  the   ver-sine  of  its  meridian  curve. 

TIIE    CENTRES     OF    GRAVITY    OF    VOLUMES. 

*  l 

§123. — When  it  is  the  question  to  determine   the  centre  of  gravity 
of  the   volume  of  any  body,  we   have 

dV  =  dx  .dy  .dz, 
and  Equations  (02)  become, 


x.  = 


/»*'      /*yr     px' 

I  ft    In    I  .t  x.dy.dz. 

«£  X         J  %.         J  2 


dx 


100  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

»xr     py'     pz' 


yr*xr     py     pz' 
"  I  n   I  tt  V -dy .  dz.dx 
x      J  y      «/  2 
Vt  —     j7 1 


y"*x'     py'     pz' 
tt   I  ,,    I  ,,  z.dy  .dz.d 
x    J  y     J  x 


x 


zt  =  : 


and, 


px'     py'     pz' 

y  —    In    I  „   In  dy  .dz.dx. 
J  x     J y     J  z 


In  which  the  triple  integral  must  be  extended  to  include  the 
entire  space  embraced  by  the  surface  of  the  body  ;  this  surface 
being   given   by  its  equation. 

If  the  volume  be  symmetrical  with  respect  to  any  line,  this  line 
may  be  assumed  as  one  of  the  co-ordinate  axes,  as  that  of  x ;  in 
which  case,  if  X  represent  the  area  of  a  section  perpendicular  to  this 
axis,  and  x,  its  distance  from  the  plane  y  z,  then  will  Xdx,  be  an 
elementary  volume  symmetrically  disposed  in  regard  to  the  axis  x, 
and  Equations  (92).  become 

px' 

I  ,f  Xxd  x 

X,      =    ~ y ,         •..••••  (107) 

yt  =  o, 


*,  =  o, 


and. 


pxf 

v  =  J  „  xdx 


(108) 


Example  1. — Find  the  position  of  the  centre  of  gravity  of  a  semi- 
ellipsoid,  the  equation  of  whose  surface  is 


X2  y2  z2 

A2^"  B2^   C2 


The    semi-axes  of  the  elliptical  section  parallel  to  the  plane  yz,  are. 


MECHANICS     OF     SOLIDS.  101 

whence, 

X=*JBc(l-5), 

and,  Equations  (107)  and    (108), 


^ =   8^ 


£*B0(i-ii)d> 


A'' 
If  the  figure  be  one  of  revolution  about  the  axis  of  x.  then,  denoting 

by 

F{xy)  =  0, (10») 

the  equation  of  the  meridian  section  by  the  plane  x  y,  will 

X=«y2, 
and  Equations  (107)  and  (108),  may  be  written, 

I  ,,    <k  y2  x  dx 

**  = y ' (ll°) 

V=  flt«yidx (Ill) 

Example  1. — Required    the  position    of  the   centre   of   gravity    of  a 
paraboloid  of  revolution. 

In  this  case,  Equation  (109), 

J>  (xy)  =  y2  —  2px  =  0, 
whence, 

V  =  2it  p  I    xd x, 

J  a 


i 


2  <v  p   I   x2  d  x        o 
x   =  • =  —a. 


2  ir  p    I    x  d  x 


3 


102  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

Exan.ph  2. — Required    the  position  of  the   centre   of  gravity   oj    tht 
volume  of  a  spherical  segment. 


whence. 


c  c 


or. 


F{xy)  =  y2  +  x*  -a*  =  0, 


V  =  «r  fX\a*  -  x*)dz 

Jx" 

*  /  /»   (a2    ~~    X2)  ,X  '  ^X 

*  /     (a2  -  s2)  a7* 


*'  =  T  'li?£*f ,—  *"2)  -*'  (3a2  -  z'»)J; 


and  for  a  segment  uf  one    base,    x"  =  a, 


JR. 


1       q*  -a:'2  ('2a2  -  x'2) 
T  '  2'/3  -  x'    (3a2  -  a:'2)' 


If  the    volume    have  a   plane  face,  and  be  of  such  figure  that    the 

areas  of  all  sections   parallel   to    this    face,  are  connected  by  any  law 

of  their  distances  from  it,  the  position  of  the  centre  of  gravity,  may 

also  be  found  by  the  method  of  single  integrals. 

< 

Example  1 . — Find  the  centre  of  gravity  of  any  pyramid. 

Find  by  the  method  explained,  the  centre  of  gravity  of  the  base 
of  the  pyramid,  and  join  this  point  with  the  vertex  by  a  straight  line. 
All  sections  parallel  to  the  base  are  similar  to  it,  and  will  be  pierced 
by  this  line  in  homologous  points  and  therefore  in  their  centres  of 
gravity.  Each  section  being  supposed  indefinitely  thin,  and  its  weight 
acting  at  its  centre  of  gravity,  the  centre  of  gravity  of  the  entire 
pyramid  will,  §  94,  be  found  somewhere  on  the  same  line. 

Take  the  origin  at  the  vertex,  draw  the  axis  x  perpendicular  to 
the    plane    of    the    base,    aria*     the    plane   xy    through   its    centre    of 


MECHANICS    OF    SOLIDS. 


103 


giavitv  ;  and    let  X  represent  any   section    parallel   to    the    base,  then 
will  Equations  (92)  become, 


£ 


Xxdx 


V 


y, 


,,  X  ydx 
_^_ _ , 

zt  =  0, 


and. 


V  =  jp  Xdx. 

Represent  by  A  the    base  of  the  pyramid,  c  its  altitude,  and   let 

y  —  ax,    . 

bo    the   equation  of  the  line  joining  the  vertex   and  centre  of  gravity 
of  the   base. 
Then, 

A  :  X  :  :  c4  :  sr*, 

#       Ax* 

A     —    r-1 


and  for   any  frustum, 


—  I"  x6  a  X 
czJ* 


'  Ax2dx 


*<=A 


c2dx 


a  A 


Vt  = 


/  n  x6 ax 

A  rxt 

I  x2d  x 


3       /*""  -  ,r^\ 


C2*> 'x 


and  for   the    entire   pyramid,  make  x"  =  c,  and  a:'  =  0,   which  give 


8 


y,  =  ?<**; 


104  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

wIhmu-"  we  conclude  that  the  centre  of  gravity  of  a  pyramid  is  on 
the  line  drawn  from  the  vertex  to  the  centre  of  gravity  of  the  base, 
and  at  a  distance  from  the  vertex  equal  to  three-fourths  of  the  length  of 
this  line. 

The  same  rule  obviously  applies  to  a  cone,  since  the  result  is  inde- 
pendent of  the  figure  of  the  base. 

The  weight  of  a  body  always  acting  at  its  centre  of  gravity,  and 
in  a  vertical  direction,  it  follows,  that  if  the  body  be  freely  sus- 
pended in  succession  from  any  two  of  its  points  by  a  perfectly 
flexible  thread,  and  the  directions  of  this  thread,  when  the  body  is  in 
equilibrio,  be  produced,  they  will  intersect  at  the  centre  of  gravity  ; 
and  hence  it  will  only  be  necessary,  in  any  particular  case,  to  deter- 
mine this  point  of  intersection,  to  find,  experimentally,  the  centre  of 
gravity  of  a  body. 

THE    CENTROBARYC    METHOD. 

§  124. — Resuming  the  second  of  Equations  (95)  and  (103),  which 
are. 


f'y  dxJ\  +d  y* 

nr.  ▼ 


in  which 


d  xl 

y,= ■ 


'=//W>  tHr 


and 


y, 


in  which 


x 

_-  > 

S 

/xl 
n(y"~y')  dx ; 


clearing   the    fractions  and   multiplying   both    members    by    2tt,  we 
shall  have, 

2n.y,s  —fl  2ny  y/dx2+dy\        ....     (112) 
ZTtyis=f*'lTT{y'2-y'2)dx     ...         .    (113) 


MECHANICS     OF     SOLIDS. 


105 


The  second  member  of  Equation  (112)  is  the  area  of  a  surface 
generated  by  the  revolution  of  a  plane  curve,  whose  extremities 
are  given  by  the  ordinates  answering  to  the  abscisses  xr  and  x", 
about  the  axis  x.  In  the  first  member,  *  is  the  entire  length  of 
this  arc,  and  2<tyt  is  the  circumference  generated  by  its  centre  of 
•rraviry.  Hence,  we  have  this  simple  rule  for  finding  the  area  of  a 
tigure    of    revolution,  viz.  : 

Multiply  the  length  of  the  generating  curve  by  the  circumference 
described  by  Jts  centre  of  gravity  about  the  axis  of  notation;  the 
product   will   be    the    required   surface. 

The  second  member  of  Equation  (113)  is  the  volume  generated 
by  a  plane  area,  bounded  by  two  branches  of  the  same  curve  or 
by  two  different  curves,  and  the  ordinates  answering  to  the  abscisses 
x'  and  x'\  about  the  axis  x.  s,  in  the  first  member,  is  the  generating 
area,  and  2tfy/  the  circumference  described  by  its  centre  of  gravity. 
Hence,  this  rule  for  finding  the  volume  of  any   figure  of  revolution,  viz. : 

Multiply  the  generating  area  by  the  circumference  described  by  its 
centre  of  gravity  about  the  axis  of  rotation ;  the  product  will  be  the 
volume   sought. 

Example  1.  —  Required  the  measure  of  the  surface  of  a  right  cone. 

Let  the  cone  be  generated  by  the 
rotation   of  the   line   A  B  about   the  ff 

line  A  C.  The  centre  of  gravity  of 
the  generatrix  is  at  its  middle  point 
G,  and  therefore,  the  radius  of  the 
circle  described  by  it  will  be  one- 
half  of  the  radius  C  B,  of  the  circu- 
lar base  of  the  cone.      Hence, 

BC.AB 


G 


2<x  yt  .  s  =  2*  . 


=  *  BC.AB. 


Example  2. — Find  the  volume  of  the  cone. 

The  area  of  the  generatrix   ABC,  is  \BC.A  C\   and  the  radius 
of  the  circle  described  by  its  centre  of  gravity  is  \BC.     Hence, 


%,9i.  =  X*BC*°4°-=.™±*± 


106 


ELEMENTS     OF     ANALYTICAL     MECHANICS 


CENTRE    OF    INERTIA. 

i 

§  125. — When  the  elementary  masses  of  a  body  exert  their  forces 
of  inertia  simultaneously  and  in  parallel  directions,  they  must  expe- 
rience equal  accelerations  or  retardations  in  the  same  time,  and  the 
factor 

cPs 

hi  the  measures  of  these  forces,  as  given  in  Equation  (13),  must  be 
the  same  for  all.  Substituting  these  measures  for  P\  P'\  &c,  in 
Equations  (62),  we  find, 


./ 


JC      33 


e   ss 


•  2ra«' 


~dP 


d2s 
IF 

~dF 


2  m 


2my' 


•  2  m 


•  2  m  zT 


~d¥ 


•  2  m 


2mx' 
2  m 


2  m    \    { 


2  rat' 
2  m 


(114) 


Whence,  Equations  (88),  the  centre  of  inertia  coincides  with  the 
centre  of  gravity  when  the  force  of  gravity  is  constant,  both  being  at  the 
centre  of  mass.  In  strictness,  however,  the  centre  of  gravity  is 
always  below  the  centre  of  inertia;  for  when  the  variation  in  the 
force  of  gravity,  arising  from  change  of  distance,  is  taken  into 
account,  the  lower  of  two  equal  masses  will  be  found  the  heavier. 
And  in  bodies  whose  linear  dimensions  bear  some  appreciable  propor- 
tion to  their  distances  from  the  centre  of  attraction,  the  distance 
between  these  centres  becomes  sensible,  and  gives  rise  to  some  curious 
phenomena. 


mechanics   of   solids, 


107 


MOTION     OF    THE    CENTRE    OF    INERTIA. 

§  126. — Substitute  in  Equations  (B),  the  values  of  c?2ar,  c?2y,  and  <Pz* 

given  by  Equations  (34),  and  we  have,  because  dt  is  constant,  and 
d2xt,  d2yt  and  d2zn  will  each  be  a  common  factor  for  all  the  elemen- 
tary masses, 


2  P  cos  a  —  M' 


V  - 


2  P  cos  (3  —  M- 


2  P  cos  y  —  M' 


dlxt 

dt2 

1 

dt2 

d*yt 

dt2 

1 

dt2 

d2zt 

1 

<^2 


dt2 


2  w.cPV  =  0, 


Zm.d2y'  =  0, 


2m.rf2z'  =  0. 


in  which  M,  denotes  the  entire  mass  of  the  body,  being  equal  to  2  m. 
Denote  by  x,  y,  z,  the  co-ordinates  of  the  centre  of  inertia  referred 
to  the  movable  origin,  then,  Equations  (114), 

M .  x  =  2  ma?', 
M.y  =  I*  my', 
M.z  =  2  m  z\ 

and  differentiating  twice, 

M .  cPx  =  2  m  .  cPx'y 

M.  cPy  =  2  m .  cPy',    y    .     .     . 

M.d*z  =  2m.  </V, 

which  substituted  in  the   preceding  Equations,  give, 


(115) 


2  P.  cos  a 

2P-C03/3 

2  P  •  cos  y 


(fir 
dP 


-  M. 


dP 


,_    <JP-x 


cPz 


dt2 


M. 


dP 
cPz 

~dfl 


^0, 


•     •     • 


(116) 


108  ELEMENTS     OF    ANALYTICAL    MECHANICS 

and  if  the   movable    origin    be    taken    at   the   centre   of  inertia,    then 
will, 

cPx  =  0,     cPy  =  0,     dh,  =  0 ; 

and   xn  yt,  ##r«  will  become  the  co-ordinates  of  the  centre  of  inertut 
referred  to  the  fixed  origin,  and  we  have, 

cPx 
2  P. cos  a  -  M>  —4-'  =  0, 

dt2 


2P.cos/3  -  M>  ^f  =  0, 

dt2 

d2z 
2  P.  cos  7  —  M •  -^  =  0; 


•     •     •     • 


(117) 


Equations  which  are  wholly  independent  of  the  relative  positions 
of  the  elementary  masses  m\  m"  &c,  since  their  co-ordinates  x\  y\ 
zf,  <fec,  do  not  enter.  It  will  also  be  observed  that  the  resistance  of 
inertia  is  the  same  as  that  of  an  equal  mass  concentrated  at  the 
body's  centre  of  inertia. 

Whence  we  conclude,  that  when  a  body  is  subjected  to  the  action 
of  any  system  of  extraneous  forces,  the  motion  of  its  centre  of  inertia 
will  be  the  same  as  though  the  entire  mass  were  concentrated  into 
that  point,  and  the  forces  applied  without  change  of  intensity  and 
direction,  directly  to  it. 

This  is  an  important  fact,  and  shows  that  in  discussing  the  motion 
of  translation  of  bodies,  we  may  confine  our  attention  to  the  motion 
of  their  centres. of  inertia  regarded  as  material  points. 

ROTATION    AROUND    THE    CENTRE    OF    INERTIA. 

§  127. — Now,  retaining  the  movable  origin  at  the  centre  of  inertia, 
substitute  in  Equations  (C),  the  values  of  d2x%  d2y,  and  d?z.  as  given 
by  Equations  (34),  and  reduce  by  the  relations, 

M.x  =  2m.x'  =  0, 


MECHANICS    OF    SOLIDS. 


09 


and  we  have, 


2  P.  (cos  /?  .  x  —  cos  a .  y') 


Sir.    —4-.  a:' — .  y  1 

fdx'     ,      dh'      A 
m,  [ .  z  — x  I 

V  df2  df2       ) 

2  P.  (cos  y  .  ?/ —  cos  (3  .z)  —  2  m  •  | y' —  — ■—.  z'  \ 


2  P.  (cos  a  . z'  —  cos  y .  x)  —  2  m 


=  0, 


=  0 


=  0 


!l 


(118) 


from  which  all  traces  of  the  position  <»f  the  centre  of  inertia  have 
disappeared,  and  from  which  we  infer  that  when  a  free  bod}T  is  acted 
upon  by  any  system  of  forces,  the  body  will  rotate  about  its  centre 
of  inertia  exactly  the  same  whether  that  centre  be  at  rest  or  in 
motion. 


§  128. — And  we  are  to  conclude,  Equations  (117)  and  (118),  that 
when  a  body  is  subjected  to  the  action  of  one  or  more  forces,  it  will 
in  general,  take  up  two  motions — one  of  translation,  and  one  of  rota- 
tion, each  being"  perfectly  independent  of  the  other. 

§  129. — Multiply  the  first  of  Equations  (117),  by  y  ,  the  second  by 
xt ,  and  subtract  the  first  product  from  the  second  ;  also,  the  first  by 
z(,  the  third  by  j?,,  and  subtract  the  second  of  these  products  from 
the  first;  also  the  third  by  yt,  and  the  second  by  st  and  subtract 
the  second  of  these  products  from  the  first,  and  we  have, 

S(Pcosj3).a?,-2(Pcosa).y,-if.  (^-*,  -"^'Vt)  =  °- 

/d2x  (Pz         \ 

S(Pcoaa).t,-l(Peoay).xl-Af-  {-ji^,  --j£".)  =  °>    '  ("») 

?(i»co8r).y,-?(J»««i/a).«J-if(5-r--^-«()  =©; 


Equations  from  whieh  may  be  found  the  circumstances  of  motion 
of  the  centre  of  inertia  about  the  fixed  origin. 


110 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


MOTION    OF    TRANSLATION". 


§  130. — Regarding  the  forces  as  applied  directly  to  the  centre  of 
inertia,  replace  in  Equations  (117),  the  values  2  P.  cos  a,  2  P.  cos  (3t 
and  2  P,  cos  y;  by  X,  F,  and  Z,  respectively,  and  we  may  write,, 


dv 


1  '  dt   ~  U' 


(120) 


from  which  the  accents  are  omitted,  and  in  which  x,  y,  and  z,  must 
he  understood  as  appertaining  to  the  centre  of  inertia 


GENERAL   THEOREM    OF    WORK,     ENERGY     AND    LIVING    FORGE. 

5  131. — Multiply  the  first  of  Equations  (120)  by  2  d  .r,  the  second 
by  2  d  y,  the  third  by  2  d  z,  add  and  integrate,  we  have 


2j'(Xdx  +  Ydy  +  Zdz)  -  M ., 


dx2  +  dy2  +  rfz2 


+  C  =    t\ 


But, 


dx2  +  rf#2  -+-  dz2 


dfi 


dt2  - 


whence. 


%f(Xdx  +  Ydy  +  Zdz)  -  M.V2+  C  =  0    •     •     (121) 


The  first  term  is,  §  101,  twice  the  quantity  of  energy  expended  or 
of  work  done  by  the  extraneous  forces,  the  second  is  twice  the  quan- 
tity of  work  of  the  inertia,  measured  by  the  living  force,  and  the  third 
is  the  constant  of  integration,  or  twice  the  quantity  of  work,  of  inertia 
in  system  before  the  forces  began  to  act. 

If  the  forces  X,  Y,  Z,  be  variable,  they  must  be  expressed  in 
functions     of     x,    y,    z,    before    the    integration    can    be    performed. 


MECHANICS    OF    SOLIDS.  Ill 

Supposing  this  latter  condition  fulfilled,  and  that  the  forms*  of  the 
functions  are  such  as  make  the  integration  possible,  we  may  write, 

F{xyz)  -  \M.V>  -f  C'=  0,      .     .     ;     .     (122) 

und  between  the  limits  x4  y4  z4    and    x/  y'  z/  , 

*W  y/  */)  -  F<*.  v,  *,)  =  i  *&*  -  7S% :   ■  (123) 

whence  we  conclude,  that  the  quantity  of  energy  expended  by  the 
extraneous  forces  impressed  upon  a  body  during  its  passage  from  one 
position  to  another,  is  equal  to  half  the  difference  of  the  living  forces 
of  the  body  at  these  two  positions. 

We  also  see,  from  Equation  (123),  that  whenever  the  body 
returns  to  any  position  it  may  have  occupied  before,  its  velocity  will 
be  the  same  as  it  was  previously  at  that  place.  Also,  that  the 
velocity,   at  any   point,   is  wholly   independent  of  the  path  described. 

If 

Xdx  +  Ydy  +  Zdz  =  Q, 

the   extraneous  forces  will,  §101,   be  in   equilibrio,  and 


v 


2.  L" 


that    is,    the    velocity   will   be    constant,    and    the    motion,  .  therefore, 
uniform, 

§  132. — Again,  multiply  the  first  of  Equations  (118)  by  rf<p,  the  sec* 
ond  by  d  t/>,  the  third  by  d  zs ;  add  and  reduce  by  the  relations  given 
in  Equations  (38) :    we  find 

,   ,  .       r,  ,   ,,       r,  .  ,  /<Pz.dz'  ,  d'y'dy-    d72.dz\- 

2  P  cos  a  dz+ZPcosfidy'+Z  P  co&ydz='zm  ( — 1 'ttt^  "i Tl — ); 

\      dt  dt  dr     F 

integrating    and    replacing   the    first    member    by  its   equal    in '  Equation 
(68),  we  have 

Denoting  the  lever  arm  of  R  by  A",  the  velocity  of  the  molecule  m  in 
reference  to  the  centre  of  inertia  by  ^*,  <fcc.,  and  the  arc  described  bv  a 


112  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

point  in  the  plane  of  the  resultant  R  and  of  its  lever  arm,  at  the  unit's 
distance  from  the  centre  of  inertia,  by  s„  we  have 

Pt>j          Pr>  rs  j           dx''  +  dy'*  +  dz'> 
J  Rdr  =J  RK.dst\ -^ =  v\  &c; 

whence 

fR.K.ds,  =  12  7??/+  C 

Adding  this  to  Equation  (121),  there  will   result 

2f(Xdx  +  Vdy  +  Zdz)  +  2j*R.K.ds,  =  AfV*+  Iwr**+C   (121/ 

From  which  it  is  apparent  that  the  quantity  of  energy  expended  upon 
a  body,  or  the  living  force  with  which  it  will  move,  is  dependent  not 
only  upon  the  intensity  of  the  force,  but  also  upon  the  distance  of  its 
line  of  direction  from  the  centre  of  inertia. 

v  §  133. — If  Equation  (121)  be  applied  to  each  one  of  a  collection  of 
elements,  of  which  the  masses  are  m,  m\  <fec,  there  will  be  as  many 
equations  as  elements ;  and  if  the  velocities  of  these  elements  be  de- 
noted by  y,  v',  &c ,  we  have,  by  addition, 

2  zf(X  d  x  +  Yd  y  +  Z  d  z)  =  2  m  v<  -  C  '  .     .     (121)" 

Let  the  extraneous  forces  be  onlv  those  arising:  from  the  mutual  actions 

and  reactions  of  the   elements   upon   one  another.      If  the  elements  m 

and  m'  be  separated  by  the  distance  r,  and   their  co-ordinates  be  xyz 

and  x'  y'  z\  respectively,  then,  the  reciprocal  action  being  along  r,  will 


x  —  x' 

fVtt  ft  —  ■            ■-■  • 

cos  /J  =  V      V  ; 
r 

z-z' 

r 

r 

,           x  —  x' 

pao  n   —                          • 

cos  f»  =       V  -  V  ; 
r 

f            *-*'. 

r 

r 

and  for  the  element  m  we  have 

Xdx  +  rdy  +  Zdz  =  p(X^^dx  +  y^y^dy  +  Z-^dz\; 

\     r  r  r  I 

for  the  element  m', 

X'dx'+V'dy'  +  Z'dz'  =  -p(X^l^dx'+y^dy'-hZ—^dz'); 

\      r  r  r  / 

and  by  addition, 

p 


i        r 


MECHANICS    OF    SOLIDts.  113 

But 

r*  =  (x-  xj  +  (y  -  y'Y  +  (z  -  z')\ 

and  differentiating, 
rdr  =  (x  -  x')d(x  -  x')  +  (y  -  y')d(y  -  y')  +  (z  -  z')  d  (z  -  *')• 

so    that    the    second    member    above    reduces   to  Pdr\    and    Equation 
<121)"  t0  2zJ*Pdr  =  2mv'-C (121)" 

If  the  elements  be  invariably  connected  during  the  motion,  the  differ- 
entials of  r  will  be  zero,  and 

2  m  i>8  =  C. 

This  is  called  the  conservation  of  living  force. 

STABLE    AND    UNSTABLE    EQUILIBRIUM. 

134. — Resuming  Equation  (123),  omitting  the  subscript  accents, 
and  bearing  in  mind  that  the  co-ordinates  refer  to  the  centre  ol 
inertia,  into  which  we  may  suppose  for  simplification  the  body  to  be 
concentrated,  we   may  write, 

$  AT  V'2  -  \MV*  =  F(x'y'z')  -  F{xyz\ 

in    which 

F(xyz)  =  f(Xdx  +  Ydy  +  Zdz\ 

and 

dF(xyz)  =  Xdx  +  Ydy  -f  Zdz. 

Now,  if  the  limits  x' y'  z'  anil  xyz   be    taken    very   near   to   eacli 
other,  then    will 

x'  =  x  +  dx\    y'  =  y  +  dy\     z'  =  z  +  dz\ 

which   substituted    above,  give 

\MV'2  -  \MV2  =  F(x  +  dx,  y  +  dy,  z  +  dz)  -  F(xyz),  ; 

and  developing   by  Taylor's  theorem, 

i        Adx  +  Bdy  -f  Cdu 
*  *  (  +  A'dx*  +  B'dy2  +  &c.  +/>, 

in    which    D   denotes    the   sum    of    the    terms    involving    the   highei 
powers   of  dx,  dy  and  dz. 


114  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

If  £  M  V 2  be   a   maximum  or   minimum,  then   will 

Adx  +  fidy  +  Cdz  =  0; (123)' 

and   since 

Adx  +  Bdy  +  Cdz  =  dF{xyz)  =  Xdx  +  Ydy  +  Zdgt 

we.  have. 

Xdx  +   Frfy  +  Zdz  =  0. 

But  when  this  condition  is  fulfilled,  the  forces  will,  Equation  (6!>), 
be  in  equilibrio ;  and  we  therefore  conclude  that  whenever  a  body 
whose  centre  of  inertia  is  acted  upon  by  forces  not  in  equilibrio, 
reaches  a  position  in  which  the  living  force  or  the  quantity  of 
work   is  a  maximum  or  minimum,  these  forces  will   be   in  equilibrio. 

And,  reciprocally,  it  may  be  said,  in  general,  that  when  the  forces 
are  in  equilibrio,  the  body  has  a  position  such  that  the  quantity  of 
work*  or  energy  will  be  a  maximum  or  minimum,  though  this  is  not 
alwttys  vtrue,  since  the  function  is  not  necessarily  either  a  maximum  or 
a  minimum  when  its  first  differential  coefficient  is  zero. 

§  135. — Equation  (123)' ,  being  satisfied,  we  have 

$M  V'2  -  \M  F2  =  ±  (A'dx*  +  B'dy*  +  &c.  +  D)  •  •  •  (124) 

The  upper  sign  answrers  to  the  case  of  a  minimum,  and  the  lower 
to  a  maximum. 

Now,  if  V  be  very  small,  and  at  the  same  time  a  maximum,  V 
must  also  be  verv  small  and  less  %han  F,  in  order  that  the  second 
member  may  be  negative ;  whence  it  appears  that  whenever  the  system 
arrives  at  a  position  in  which  the  living  force  or  quantity  of  work  ij» 
a  maximum  and  the  system  in  a  state  bordering  on  rest,  it  cannot 
depart  far  from  this  position  if  subjected  alone  to  the  forces  which 
l)t ought  it  tnere.  This  position,  which  we  have  seen  is  one  of  equi- 
librium, is  called  a  position  of  stable  equilibrium.  In  fact,  the  quantity 
of  work  immediately  succeeding  the  position  in  question  becoming 
negative,  shows  that  the  projection  of  the  virtual  velocity  is  negative, 
and  therefore  that  it  is  described  in  opposition  to  the  resultant  of  the 
forces,  which,  as  soon  as  it  overcomes  the  living  force  already  existing, 
will  cause  the  body  to  retrace  its  course. 


MECHANICS    OF    SOLIDS.  115 

If,  on  the  contrary,  the  body  reach  a  position  in  which  the  living 
force  is  a  minimum*  the  upper  sign  in  Equation  (124)  must  be  taken, 
the  second  member  will  always  be  positive  and  there  will  be  no  limit 
to  the  increase  of  V.  The  body  may  therefore  depart  further  and 
further  from  this  position,  however  small  V  may  be;  and  hence,  this 
is  called  a  position  of  unstable  equilibrium. 

If  the  entire  second  member  of   Equation  (124)  be  zero,  then  will 

\M  V'2  -  \MV*  =  0, 

ana  there  will  be  neither  increase  nor  diminution  of  quantity  of  work 
and  whatever  position  the  body  occupies  the  forces  will  be  in  equili 
brio.     This  is   called   equilibrium  of  indifference. 

g  136. — If  the  system  consist  of  the  union  of  several  bodies  acted 
upon  only  by  the  force  of  gravity,  the  forces  become  the  weights 
of  the  bodies  which,  being  proportional  to  their  masses,  will  be  con- 
stant. Denoting  these  weights  by  W,  W",  W"\  &c,  and  assum- 
ing the  axis  of  z  vertical,  we  have  from  Equations  (87), 

Rz,  =  W'z'  +  W"  z"  4  WT"V"  +  &c, 

in  which  R,  is  the  weight  of  the  entire  system,  and  zt  the  co-ordi- 
nate of  its  centre  of  gravity;    and  differentiating, 

Rdz,  =  W'dz  +  W'dz"  4-  W"'dz'"  +  &c    .     .     .    (125) 

Now,  if  z,  be  a  maximum  or  minimum,  then  will 

W'dz'  4   W'dz"  +  W'"dz'"  4  &c.  =  0, 

which  is  the  condition  of  equilibrium  of  the  weights.  Whence,  wo 
conclude  that  when  the  centre  of  gravity  of  the  system  is  at  the 
highest  or  lowest  point,  the  system  will  be  in  equilibrio. 

In  ordei  that  the  virtual  moment  of  a  weight  may  be  positive, 
vertical  distances,  when  estimated  downwards,  must  be  regarded  as 
positive.  This  will  make  the  second  differential  of  zt ,  positive  at 
the  limit  of  the  highest,  and  negative  at  the  limit  of  the  lowest 
point.  The  equilibrium  will,  therefore,  be  stable  when  the  centre  of 
gravity  is  at  the  lowest,  and  unstable  when  at  the   highest  point. 


116      ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Integrating-  Equation   (125)  between  the  limits  z=zH  and  z  =JT, 
•  e'  =  h    and  z'  s=  h\  «fcc.,  and  we  find, 

J2  (#,  -  H)  =  IT  ft  -  h')  +  WP»  ft  -  A")  +  cfcc. ;  .     .     (126) 

from  which  we  see  that  the  work  of  the  entire  weight  of  the  system, 
acting  at  its  centre  of  gravity,  is  equal  to  the  sum  of  the  quantities 
of  work  of  the  component  weights,  which  descend  diminished  by  the 
sum  of  the  quantities  of  work  of  those  which  ascend. 

THE     POTENTIAL     FUNCTION. 

§  .137. — If,  for  any  limited  system  of  invariable  masses,  2m,  acted 
upon  by  the  forces  X,  Y,  Z,  functions  of  the  masses  and  of  the 
variables  of  position,  x,  y,  z,  integration  be  possible,  and  we  denote 
by  the  letter  IT  the  function  which  expresses  the  quantity  of  action 
of  the  forces  exerted  during  any  change  of  position,  or  configuration, 
then  will  dU  =  iPdp  =  Xdx  -f  Ydy  -f  Zdz 

be  an  exact  differential. 

This  function,  called  by  Lagrange  simply  the  function  IT,  but  now 
generally  known  as  the  potential,  is  one  of  great  importance  in  the 
investigation  of  the  forces  of  nature. 

In  Articles  131  and  134,  we  have  already  discussed  some  of  the 
relations  of  the  function, 

F(x,  y,  z,)  =  f(Xdx  +  Ydy  +  Zdz). 

But  as  the  second  member  of  this  equation  is  often  used  to  signify 
work  done,  or  resistance  overcome,  it  becomes  necessary  that  we  should 
consider  what  other  important  relations  present  themselves  when  it  is 
used  to  denote  not  work  itself,  but  the  quantity  of  action  of  the 
forces  expended  in  doing  that  work.  The  distinction  is  that  of  action 
and  reaction,  cause  and  effect;  and  we  have  already  remarked 
upon  it. 

To  indicate  that  amount  of  stored  action,  rather  than  work,  is  the 
meaning  of  the  function  II.  The  name  potential  energy  is  used  for 
it  by  Thomson  and  Rankine  ;  and  these  able  writers  also  employ  the 
terms  actual,  or  kinetic,  energy,  in  place  of  the  unmeaning  half  vis 
viva,  or  living  force,  which  measures  the  quantity  of  working  power 
of  a  moving  body.     New  names,  which  appear  to  be  coming  rapidly 


MECHANICS    OF    SOLIDS.  117 

into  general  use,  and  with  advantage  of  greater  clearness  of  thought 
and  expression. 

For  the  sake  of  brevity,  we  shall  use  the  simple  name  potential 
for  the  function  IT,  but  it  will  always  be  understood  to  mean  poten- 
tial energy,  the  power  to  do,  instead  of  the  work  done.  And  it  is 
the  quantity  of  action  of  the  forces,  attractive  or  repellent,  when 
exerted  to  produce  changes  of  position,  or  distance,  between  masses, 
or  molecules. 

To  illustrate  physically  the  meaning  of  the  term,  the  potential  of 
elasticity  for  a  bent  steel  spring  is  its  power  or  quantity  of  action 
in  the  recoil  to  a  neutral  state  of  equilibrium,  from  which  it  has  been 
forced  by  bending.  The  stored  power  of  an  elevated  weight  to  per- 
form by  descent  an  amount  of  work  is  its  potential  of  gravity.  The 
amount  of  power  exerted  by  the  iron  rim,  or  tire,  of  a  carriage 
wheel,  when  chilled  from  a  hot  state,  and  thus  made  to  contract 
powerfully  upon  the  wooden  frame  of  the  wheel,  is  its  heat  potential. 
All  such  are  simply  changes  of  power,  varying  with  relative  position; 
and  many  like  examples  might  easily  be  ~ adduced, 

i 

CONSERVATION     OF     ENERGY. 

§  138. — Power  and  work,  action  and  reaction,  always  bear  to  each 
other  the  algebraic  relation  of  positive  and  negative  quantities.  When 
forces  work,  the  potential  is,  therefore,  a  decreasing  and  the  work  an 
increasing  function.  Or  expenditure  of  power  produces  increase  of 
work.  Bearing  this  in  mind,  and  integrating  between  limits,  the 
expression 

gives 


n0-n  =  2^(,2-v). 


From  which  we  obtain, 


Hence,  it  appears  that  the  total  amount  of  power,  or  of  energy,  in 
any  limited  system  of  masses  and  forces,  in  which  the  forces  are 
functions  only  of  the  variables  x,  y,  ar,  must  always  be  constant. 


118      ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Or,  following  Lagrange,  we  may  denote  the  total  energy  of  the 
system  by  the  constant  H,  and  the  kinetic  energy,  or  half  vis  viva, 
by  F,  which  substitution  gives 

n  +  V  =  H, (D) 

an  equation  which  may  be  enunciated  thus:  the  total  energy,  both 
potential  and  kinetic,  is  constant. 

This  principle  is  now  usually  called  the  law  of  conservation  of 
energy ;  but  in  the  precise  form  here  given,  Equation  (Z>),  it  is  used 
by  Lagrange,  though  by  him  called  the  principle  of  living  force. 

It  is  evident  that  the  total  energy  of  any  system  can  be  constant 
only  when  neither  the  masses,  nor  the  forces  acting  upon  them,  all 
other  things  being  equal,  vary  with  time.  In  other  words,  there  must 
be  neither  growth  nor  decay,  neither  invigoration  nor  enfeeblement, 
with  age  or  lapse  of  time.  Any  system  thus  invariable  in  energy  is 
said  to  be  dynamically  conservative. 

If,  however,  the  energy  be  not  confined  in  the  system  itself,  but 
be  expended  upon  bodies  foreign  thereto,  then  clearly  there  will  be 
loss,  or  dissipation  of  energy.  Thus,  motive  power  in  machines  is  by 
friction  transformed  into  heat,  and  then  dissipated  by  conduction  and 
radiation.  But  if  dissipated  and  transformed,  it  is  only  transferred 
to  other  masses  and  not  annihilated;  power,  therefore,  appears  to  be 
indestructible. 

From  the  above  it  will  be  seen  that  Equation  (Z))  is  substantially 
identical  with  our  fundamental  formula  (yl),  they  being  only  equiva- 
lent algebraic  transformations  for  the  same  general  law 

DISCUSSION     OP    THE     FUNCTION     II. 

§  139. — The  potential  II  has  many  properties  which  render  it  of 
great  importance  in  the  investigation  of  the  phenomena  of  gravitation, 
elasticity,  heat,  electricity,  and  magnetism,  only  a  few  of  which  can 
here  be  given. 

When  the  action  of  the  component  forces,  X,  J",  Z,  depends 
upon  changes  of  position,  ar,  y,  z,  and  may  be  expressed  by  the  sin- 
gle potential  function 

d  II  =  Xdx  +  Ydy  +  Zdz, 


MECHANICS    OF    SOLIDS  119 

or  by  its  equivalent, 

ail  =  -—  ax  4-  -— rt?/  H — —  dz, 

dx        ^  dy    J  ^  dz      '     . 

the  requisite  conditions  of  integrability  are 

dX  _dY       dX  _dZ        dY  _dZ 
dy         dx1         dz         dx'         dz         dy' 

From  the  above  equations,  and  from  Equations  (43),  we  obtain, 

A  =  —r-  =  It  cos  a, 
</£ 

rfll 
1   =  -r—  =  it  cos  6, 
ay 

Z  s±  -t—  =s  /c  COS  c. 
dz 

And,  for  the  resultant  force, 

DO     /rfn\a     /rfn\a     /rfn\» 

Also,  for  any  component,  P,  of  this  resultant,  making  with  it  an 
angle  0,  and  acting  in  the  direction  *,  which  makes  the  angles  a,  )3,  y, 
with  the  rectangular  co-ordinate  axes, 

P  =  ficosd, 
cos  0  ==  cos  a  cos  «  -f-  cos 6  cos/3  -f-  cos  c  cosy. 

Which  values,  with  the  relations, 

dx  dy  dz 

cos«  =  — ,  cosp  =  — -,  cosy  =  — , 

ds  ds  ds 

give 

_,       dYl     dx       dll    dii       dll    dz 

P  = § j t  JL  _j t m 

dx  '  ds        dy  '  ds        dz     ds' 

But,  the  second  member  of  this  last  expression,  being  only  the  sum 
of  the  partial  derivatives,  is  equal  to  the  total  derivative  of  II,  taken 
as  a  function  of  s\   and,  therefore, 

ds' 
So  that,  for  any  direction  s,  the  component  force  is  measured  by  the 
first  differential  coefficient  of  the  potential  as  a  function  of  the  linear 
path  in  that  direction ;  a  general  relation,  precisely  analogous  to 
those  for  the  components  in  the  directions  of  the  co-ordinate  axes, 
and  of  which,  in  fact,  they  are  only  particular  cases. 


120        ELEMENTS    OF    ANALYTICAL    MECHANICS. 
For  any  constant  value  of  the  potential, 

n  =  a, 

the  equation  of  a  level,  or  equipotential  surface,  we  have 

Xdx  +  Ydy  +  Zdz  =  0] 
and  tins  divided  by  a  common  factor  becomes 

X    dx        Y  dy       Z  dz 
R'ds+Rtk+Ti  d*  ~    ' 
or,  by  substitution, 

cos  a  .  cos  a  +  cos  b  .  cos  (3  -\-  cos  c  .  cos  y  =  0. 

The  first  member  of  which  being  equal  to  the  cosine  of  the  angle  0, 
made  by  the  force  R  with  ds,  an  element  of  the  surface,  it  follows 
that  0  must  always  be  90°,  or  the  force  is  at  all  points  normal  to 
the  surface.  A  particle  thus  acted  upon  cannot  have  any  tendency 
to  move  on  the  surface,  and  must  be  (as  will  be  shown  in  §  259) 
precisely  in  the  dynamical  state  of  every  one  of  the  particles  forming 
the  level  surface  of  a  mass  of  water  acted  upon  by  the  force  of 
gravity.  Hence,  the  term  level  surfaces  is  used  for  all  those  whose 
potentials  are  constant. 

Anv    level   surface,    whose    very    snail    distance   from   the   first   is 
denoted  by  dr,  will  have  for  its  constant  potential 

n  ^=  A  +  dll ; 
i:i  which  the  value  of  the  constant  increment  is 

dr 

Bat  the  coefficient  of  6r  in  this  formula  has  been  shown  to  be  the 
component  of  the  force  R  for  the  normal  direction  oV,  and  the  force 
R  itself  to  act  in  that  direction  ;  the  normal  component  is,  therefore, 
identical  with  the  force,  and 

R  —  dU  —  ^5 
dr         dr' 

Or  the  force  varies  directly  as  the  increment  of  the  potential  and 
inversely  as  the  distance  between  the  level  surfaces. 

'And,    therefore,    an    indefinite    number   of    similar    level    surfaces, 
arranged    consecutively,    and    differing   from    each    other   only   in    the 


MECHANICS    OF    SOLIDS.  121 

respective  values  of  their  constant  potentials,  wiH  give  for  any  system 
the  values  and  directions  of  the  resultant  force  at  all  points  in  space. 

It  is  evident  that  no  two  level  surfaces  belonging  to  the  same 
system  can  either  intersect,  or  touch,  each  other ;  for  in  such  case, 
the  distance  between  them  being  zero,  the  force  would  become 
infinite,  which  is  physically  impossible. 

Also,  as  no  finite  system  of   physical  masses  and  forces  can  exert 
infinite  power,  and  all   such  forces  become  zero    at   infinite  distances, 
every    level    surface    is   a   finite    closed    surface    enveloping   the    space 
'within. 

The  potential  being  a  function  of  the  co-ordinates  of  position 
only,  it  follows  that,  if  the  forces  of  a  system  cause  any  of  its 
masses  to  depart  from  one  situation  and  go  to  another,  the  variation 
of  power  due  to  such  a  definite  change  of  position  must  always  be 
the  same  independently  of  t!:e  paths  followed.  And  if  the  ma:«oes 
should  return  again  to  their  first  positions,  then  the  power  will  be 
restored  to  its  original  value.  If  this  were  otherwise,  a  body  by 
always  going  by  one  path  of  less  potential  ehange,  and  returning  by 
one  of  greater,  could  constantly  increase  its  power,  and  that  even  to 
an  infinite  degree ;  or,  in  other  words,  perpetual  motion  would  be 
possible. 

In  x\rticles  134  and  135,  stable  and  unstable  equilibrium  are 
shown  to  depend  upon  maxima  and  minima  values  of  the  function 
which  is  the  integral  of 

Xdx  -f  Ydy  +  Zdz ; 
that  function  is  therefore   virtually  the  same  as  the  potential  function 
here   discussed.     And   all    level    surfaces    contain    maxima  or    minima 
curves. 

It  may  be  well  to  remark  that  some  writers  restrict  the  term 
potential  to  particular  cases,  when  the  forces  vary  inversely  as  the 
square  of  the  distance  and  act  upon  the  unit  of  mass.  And,  with 
Hamilton,  they  call  the  general  function  II  the  function  of  force. 
This  restriction  seems  injudicious,  not  only  because  it  excludes 
molecular  forces,  but  also  for  the  reason  that 

n  =  x/Pdp 

does   not   express   an    instantaneous   pressure  or  force,  but  rather   the 


122       ELEMENTS    OF    ANALYTICAL    MECHANICS. 

total  amount  of  work  done,  or  power  used,  in  the  continued  applica- 
tion and  expenditure  of  that  force  along  a  certain  path  of  resistance. 
The  above  are  a  few  of  the  many  important  properties  of  the 
potential  function  II ;  the  chief  value  of  which  function  is  that  it 
very  greatly  simplifies  investigation,  whenever 

Xdx  4-  Ydy  +  Zdz 
is  an  exact  differential  of  the  variables  x,  y,  z,  and  therefore  capable 
of  integration  under  the  form  of  a  single  expression.  And  this  may 
readily  be  shown,  as  in  §  133,  to  be  always  the  case  for  the  known 
physical  forces.  Full  discussion  of  the  subject  belongs,  however, 
rather  to  physics  than  to  mechanics. 

INITIAL     CONDITIONS,     DIRECT     AND     INVERSE     PROBLEM. 

§  140. — By  integrating  each  of  Equations  (120)  twice,  we  obtain 
three  equations  involving  four  variables,  viz.:  x,  y,  z,  and  t.  By 
eliminating  £,  there  will  result  two  equations  between  the  variables 
.r,  y,  and  z,  which  will  be  the  equations  of  the  path  described  by 
the  centre  of  inertia  of  the  body. 

In  the  course  of  integration,  six  arbitrary  constants  will  be  intro- 
duced, whose  values  are  determined  by  the  initial  circumstances  of 
the  motion.  By  the  term  initial  is  meant  the  epoch  from  which  t 
is  estimated. 

The  initial  elements  are,  1st.  The  three  co-ordinates  which  give 
the  position  of  the  centre  of  inertia  at  the  epoch;  and  2d.  The  com- 
ponent velocities  in  the  direction  of  the  three  axes  at  the  same  instant. 

The  general  integrals  determine  the  nature  only,  and  not  the 
dimensions  of  the  path. 

§  141. — Now  twyo  distinct  propositions  may  arise.  Either  it  may 
be  required  to  find  the  path  from  given  initial  conditions,,  or  to  find 
the  initial  conditions  necessary  to  describe  a  given  path. 

In  the  first  case,  by  integrating  Eqs.  (120)  twice,  we  obtain  six  equa- 

dx  dij  dz 

tions  in  x,  y,  2,  f,  the  component  velocities,  — ,  -^,  — ,  and  six  arbi- 

dt    dt    dt 

trary  constants  of  integration.     Making  in  these  equations  t  =  0,  and 

substituting  for  the  co-ordinates  and  component  velocities  their  initial 


MECHANICS    OF    SOLIDS. 


123 


rallies,  the  constants  become  known.  These,  in  the  throe  equations 
obtained  from  last  integration,  give  three  equations  in  #,  y,  z  and  t.  from 
which,  if  t  be  eliminated,  two  equations  in  x,  y  and  z,  will  result.  These  will 
be  the  equations  of  the  path,  and  the  problem  will  be  completely  solved. 
In  the  second  case,  the  two  equation*  of  the  path  being  differentiated 
twice  and  divided  each  time  by  dt,  give  oidy  four  equations  involving 
three  first,  and  three  second  differential  co-efficients.  The  inverse  problem 
is,  therefore,  indeterminate. 

,    But    Equation    (121)  being    differentiated    and   divided   by  the   dif- 
ferential   of  one   of  the    variables,  say  d  x,  gives 


_  _  d  V2  ■„         ,  _   d  y         „d  z 

hM-  -r-  =  X  +   Y'  T-  +  Z  T 
d  x  dx  dx 


(127) 


which  is  a  fifth  equation  involving  X,  FJ  Z,  and  V.  By  assuming 
a  value  for  any  one  of  these  four  quantities,  or  any  condition  con- 
necting them,  the  other    five    may  be  found  in  terms  of  .r,  y  aiif.   z. 


VERTICAL   MOTION   OF   HEAVY   BODIES. 

§  142. — When  a  body  is  abandoned  to  itself,  it  falls  toward  the 
earth's  surface.  To  find  the  circumstances  of  motion,  resume  Equa- 
tions (120),  in  which  the  only  force  acting,  neglecting  the  resistance 
of  the  air,  will  be  the  weight  =  My ;  and  we  shall  have,  Equa- 
tions (117), 

2  P  cos  a  =  X  =  Mg  .  cos  a  ; 

2  P  cos  (3  =  Y  —  Mg  .  cos  (3  ; 
2  P  cos  y  =  Z  =  Mg  .  cos  y ; 

in  which  M  denotes  the  mass  of 
the  body.  The  force  of  gravity 
varies  inversely  as  the  square  of 
the  distance  from  the  centre  of 
the  earth,  but  within  moderate 
limits  may  be  considered  invaria- 
ble. The  weight  will  therefore  be 
constant  during  the  fall. 

Take  the  co-ordinate  z  vertical, 
wud  positive  when  estimated  downwards,  then  will 

cos  a  as  0  ;     cos  /3  =s  0  j     cos  y  =  1, 


124  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and  Equations  (120)  become,  after  omitting   the  common   factor  M, 

d2x         _       d2y         _       cPz 

=  9, 


t 

df 

-"'     rf<«   -"'      dP 

and   integrating 

dx                dy 

dt       »;'  rf/       r* 

—    =    V    =r   <7  /  4-   W 

•  ••••• 


(128) 


in  which  v  is   the  actual  velocity  in   a  vertical   direction. 
Making  t  =  0,  we    have 

dz 

—  =  «■• 

d*  • 

The   constants    u  ,  w    and   u  ,    are    the    initial     velocities    in    the 

ar       y  z1 

directions    of  the    axes  x,  y  and   2,  respectively.     Supposing    the  first 
two   zero,  and    omitting   the    subscript   2,  from   the    third,  we    have, 

dx  dy 

dt  '    dt  ' 

c?z 
v  =  —  =  yt-\-u (129) 

(I    V 

Integrating  again,  we  find 

x  =  C;    y  r=  C, 

and  if   when    £  =  0,  the   body  be   on    the  axis   2,  and   at   a   distance 
below    the   origin    equal    to   a,    then    will 

*  =  0  ;     y  =  0  ; 
z  =  \gt2  +  ut  +  a (130) 

If  the   body  had  been   moving  upwards  at   the   epoch,  then  would 
u   have    been    negative,  and,  Equations  (129)  and  (130), 

v  —  gt  —  u *•     •     (131) 

z  =  \yt2  —  ut  +  a {132) 


MECHANICS    OF    SOLIDS.  125 

If  the  body  had  moved  from  rest  at  the  epoch  and  from  the 
origin  of  co  ordinates,  then  would  v  be  the  actual  velocity  generated 
by  the  body's  weight,  and  z  =  h,  the  actual  space  described  in  the 
time  t\  and  Equations  (129)   and  (130)  would  become, 

v=gt (133) 

h  =-kfft2. (131) 

and  eliminating  /, 

v  as  yfWfh (135) 

whence,  we  see  that  the  velocity  varies  as  the  time  in  which  it  is 
generated  ;  that  the  height  fallen  through  varies  as  the  square  of  the 
time  of  fall ;  and  that  the  velocity  varies  directly  as  the  square  root 
of  the  height. 

The  value  of  A,  is  called  the  height  due  to  the  velocity  v  ;  and 
the  value    «/,  is  called  the  velocity  due  to  the  height  h. 

If,  in  Equation  (132),  we  suppose  a  =  0,  we  shall  have  the  case 
of  a  body  thrown  vertically  upwards  with  a  velocity  u.  from  the 
origin,  and  we  may  write, 

v  =  gt-u, (136) 

I  =  \gfl  —  ut\ (137) 

when  the  body  has  reached  its  highest  point,  v  will  be  zero,  and  we 
find, 

g  t  —  u  =  0 ; 
or. 


> 


u 
t  -  —i 

9 


which  is  the  time  of  ascent;    and  this  value  of  t,  in  Equation  (137), 
will  give  the  greatest  height,  h  =  z,  to  which  the  body  will  attain, 


«2 


§143. — In    the   preceding   discussion,    no   account   is   taken    of    the 
atmospheric  resistance       For  the  same  body,  this  resistance  varies  m 


126  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

the  square  of  the  velocity,  so  that  if  £,  denote  the  velocity  when  the 
resistance  becomes  equal  to  the  body's  weight,  then  will 

M  .  g  .  v2 


be  the  resistance   when    the   velocity  is  v,  and  in  Equations  (117),  we 
shall  have, 

v2 

2  P  cos  a  ss  X  =  M  g  cos  a  -\-  M  g  -  —  •  cos  a', 

K 
*  2 

2  P  cos  /3  =    F  =  if  a  COS  /3  -f  if  flr  .         .  Cos  /3', 
2  P  cos  7  =  Z  =  M g  cos  7  -f-  M g  •  —  •  cos  7' ; 

taking  the  co-ordinate  z,  vertical  and  positive  downward,  then  will, 

cos  a  ss  cos  a!  =  0, 
cos  (3  =  cos  /S'  =  0, 
cos  y  s=  1,    cos  7'  =  —  1  ; 

and  Equations  (120)  give, 

1 

c?2  z         .  rfv 

Omitting  the  common  factor  if,  and  replacing  — —  by  its  value  —  - 

dv  /  v2  \ 

whence, 

F.rfv  &  (dv  dv     \ 

integrating  and  supposing  the  initial  velocity  zero, 

!,t  =  ik.log    £4—-      ...         •     (140) 


MECHANICS    OF    SOLIDS.  127 

which  gives  the  time  in  terms  of  the  velocity;  or  reciprocally, 


r> 


in    which  e,  is    the  base  of  the    Naperian    system  of  logarithms,  and. 
from  which  we  find, 


e  k  —  e     k  ) 


v  =  — - 

e  k   +  t    k 

which    gives   the   velocity    in    terms  of  the   time.     Substituting  for  », 

dz 
its  value   —  i      integrating   and    supposing    the    initial    space   zero,    w« 

(X  v 

have 

k2               ,  —         -'—^ 
2  =  l.logi^-    +e    A (143j 

Multiplying   Equation   (139)   by 

dz 


we   have, 


5i=  •• 


.  k2.v  .dv 

adz  —  1 


and    integrating,  observing   the   initial    conditions   as   above, 

k2  k2 

which   gives    the    relation   between  the    space   and    velocity. 

_  l± 
As    the   time    increases,  the    quantity  e     *    becomes  less  and  less, 

and    the   velocity,    Equation    (142),    becomes    more    nearly    uniform  ; 

for,  if  t  be   infinite,  then  will 

r  *  =  o, 

and,    Equation    (142), 

v  =  k\ 

making    the    resistance   of  tho   air    equal    to    the   body's    weight. 


128  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

§144. — If    the   body    had    been   moving   upwards   with   a   velocity 
v,    then,  taking   z   positive    upwards,    would,   Equations    (120), 

d2  z  a  v2 

dv  d2  z 

substituting   — -    for   -ry   an^   omitting  the  common  factor,  we  find, 


k  .dv  g  d  t 


k2  +  v2  k    ' 

integrating, 

~~l  v  9  *        ^ 

tan    T  =  -T+C; 

and   supposing   the   initial  velocity  equal  to  a,  we  find 

_1  « 
V  =  tan     — > 

k 


(145) 


tan     -  =  tan     —  -  *— (146) 


nid, 

—l 

tan 

k  k  k 

Taking  tne   tangent   of  both  members  and  reducing,  we  find 

fft 
a  —  k .  tan  -— ■ 

t  =  * (147) 

at 

k  4*  «  •  tan  — 
k 

which  may  be  put  under  the  form, 

9*        j     .    9* 
a .  cos  — k .  sin  — 

vz=k 1 1      ....     (148) 

•    9*    ,    .  9*  . 

a  .  sin  - — (-  *  •  cos  — 
&  At 

Substituting    for  v   its  value   — >    integrating,    and    supposing    the 
initial  space  zero,  we  have 

k2      i       (  a        •     9*  9*\  fiAQ\ 

f  s=  —  •  log  (  —  •  sin  V  4-  cos  T)        •  •    (I4y) 

g         *  \  k  k  k  / 


MECHANICS    OF    SOLIDS.  129 

Multiplying  Equation  (145),  by 

dz 

and  we  have, 

k2  .v.dv 


g  ,dz  =  — 


k2  +  v2   ' 


and  integrating,  with  the  same  initial  conditions  of  v   being   equal    to 
a,  when  z  is  zero,  there  will  result, 

«  =  -*!  .  log   F  +  a2 (150) 

2g        &   k2  +  v2  v       ; 

§145. — If  we  denote  by  A,  the   greatest  height  to  which  the  body 
will  ascend,  we  have  z  =s  h,  when  v  —  0,  and  hence, 

h  =  Jl..}0g   F  +  a2 (151) 

2g        ^        k2  y       ' 

Finding  the  value  of  £,  from  Equation  (146),  we  have, 

-i   n  -i 


t 


=        (tan      T-tan       y)    .     .     .     .     (152) 


from  which,  by  making  v  =  0,  we  have, 

k  — l  a  .  __. 

*o  = tan      —       .......     (153) 

9  * 

which  is  the  time  required  for  the  body  to  attain  the  greatest  eleva- 
tion. Having  attained  the  greatest  height,  the  body  will  descend,  and 
the  circumstances  of  the  fall  will  be  given  by  the  Equations  of  §  143. 
Denoting  by  a',  the  velocity  when  the  body  returns  to  the  point  of 
starting,  Equation  (144),  gives, 

k2     .  K* 

h  =  irr  '  ]°g 


2g         &   k2  -  a'2 

and  placing  this  value  of  h  equal   to    that   given    by  Equation  (151  ^ 
there  will  result, 

k2  k2  +  a2 

-* 


k2  -  a'2  k2 


130  ELEMENTS     OF     ANALYTICAL    MECHANICS 

whence, 


a*  =  a2 


a2  +  k2   ' 


that    is,  the  velocity  of   the  body  when    it   returns    to    the    point    of 
departure  is  less  than  that  with  which  it  set  out. 
Making  v  =  af  in  Equation  (140),  we  have, 

and,  substituting  for  a',  its  value  above, 


(154) 


k       .       %/  a2  -f  k2  +  a 
t .  =  — —  •  log  — ,  •     •     .     . 

a  value  very  different  from  that  of  ta,   given  by  Equation   (158),  for 
the  ascent. 

Multiplying  both  numerator  and  denominator  of  the  quantity  whose 
logarithm  is  taken,  by   v/«2  +  ^  —  a,    the  above  becomes, 

•  *'  =  f  ,og7PW^ (,55) 

Adding  Equations  (153)  and  (155),  we  have, 

k  T      -1  a  k  "I 

t    +  t,  =  —     tan      —  -f  log 

n%  making  t  =  ta  +  t- 

T  =  tan    T  +  log  7^T=f '   "   "   (156) 

K  K  y  k2  -j-  a2  —  a 

If  a  ball  be  thrown  vertically  upwards,  and  the  time  of  its 
absence  from  the  surface  of  the  earth  be  carefull  v  noted,  t  will  be 
known,  and  the  value  of  k  may  be  found  from  this  equation.  This 
experiment  being  repeated  with  balls  of  different  diameters,  and  the 
resulting  values  of  k  calculated,  the  resistance  of  the  air,  for  any 
given  velocity,  will  be  known.  i 


MECHANICS    OF    SOLIDS 


131 


PROJECTILES. 

§  146. — Any  body  projected  or  impelled  forward,  is  called  a  pro- 
jectile, and  the  curve  described  by  its  centre  of  inertia,  is  called  a 
trajectory.  The  projectiles  of  artillery,  which  are  usually  thrown  with 
great  velocity,  will  be  here  discussed. 

§  147. — And  first,  let  us  consider  what  the  trajectory  would  be 
in  the  absence  of  the  atmosphere.  In  this  case,  the  only  force  which 
acts  upon  the  projectile  after  it  leaves  the  cannon,  is  its  own  weight ; 
and,  Equations  (1 17), 

2  P  cos  a  =  X  =  Mg  cos  a, 
2  P  cos  £  =  Y  —  Mg  cos  (3, 
2  P  cos  y  =   Z  =  M g  cos  y. 


Assuming  the  origin 
at  the  point  of  de- 
parture, or  the  mouth 
of  the  piece,  and 
taking  the  axis  z 
vertical,  and  posi- 
tive upwards,  then 
will 


cos  a  =  0 ;    cos  (3  =  0;    cos  j  =  —  1  ;    and,  Equations  (120), 
M~=0;    M-d-JL=0-    M>~=  -Mg; 


dt2 


dt2 


dt2 


and  integrating,  omitting  if, 


d  x 
~d~t 


=  u 


dy 

dt 


d  z 


=  u   ;     -7 —  =  —  gt  -j-  u 

y '  .  dt  y  * 


(157) 


Integrating  again,  and  recollecting  that  the  initial  spaces  are  zero,  we 
have, 


x  — 


■=  u   •  t ;    y  =  u    •  t  ;        =•.  —  £  g  -2  -f-  u    •  t     • 


132  ELEMENTS     OF    ANALYTICAL    MECHANICS 

and  eliminating  /,  from  the  first  two,  we  obtain, 


u 

z 


wnich    is    the    equation  of  a   right   line,  and    from  which  we  see  that 

the  traje3tory  is  a  plane  curve,  and  that  its  plane  is  vertical. 

Assume  the  plane  zx,  in  this  plane,  then  will  y  =  0,  and  Equa- 
tions (158),  become, 

x  =  ux  •  t  ;    z  —  —  \g  t2  -|-  *    -  t      •     •     •      (159) 

Denote  by  V,  the  velocity  with  which  the  ball  leaves  the  piece, 
that  is,  the  initial  velocity,  and  by  a,  the  angle  which  the  axis  of  the 
piece  makes  with  the  axis  a?,  then  will, 

V.  cos  a,    and     V  .  sin  a, 

be  the  lengths  of  the  paths  described  in  a  unit  of  time,  in  the  direc- 
tion of  the  axes  x  and  z,  respectively,  in  virtue  of  the  velocity  V  ; 
they  are,  therefore,  the  initial  velocities  in  the  directions  of  these 
axes;   and  we  have,  , 

u    =  V  cos  a  :     u    =  V .  sin  a  ; 

X  '  X 

which,  in  Equations  (159),  give 

x  =  V .  cos  a .  t ;     z  =  -  £  g  t2  -f  V.  sin  a  .  t  •     (160) 

and   eliminating  t,  we  find 

Q     X2 

z  =  x  tan  a  > 


2  V2 .  cos2  a  ' 
or   substituting  for   V  its  value   in  Equation  (135), 


x2 


z  =  x  tan  a  —  — -—  •     •     •     (161) 

4  h .  cos*  a 


which  is  the  equation  of  a  parabola. 


MECHANICS     OF     SOLIDS 


133 


§  148. — The  angle  a  is 
called  the  angle  of  projec- 
tion ;  and  the  horizontal 
distance  A  D,  from  the 
place  of  departure  A,  to 
the  point  Z>,  at  which  the 
projectile  attains  the  same 
level,  is  called  the  range. 

To  find  the  range,  ma«ve  z  =  0,  and  Equation  (161)  gives 

x  =  0,    and    x  =  4  k  sin  a  cos  a  =  2  h  sin  2  a, 


and   denoting   the   range   by  i?, 


i2  =  2  ^ .  sin  2  a 


(162) 


the    value    of  which   becomes   the   greatest    possible   when   the   angle 
of  projection    is   45°.     Making  a  =  45°,  we   have 


R  =  2k 


(163) 


that   is,    the    maximum     range    is   equal    to    twice    the   height  due   to 
the   velocity  of  projection. 

From  the  expression  for  its  value,  we  also  see  that  the  same 
range  will  result  from  two  different  angles  of  projection,  one  of  which 
is   the   complement  of  the   other. 


149. — Denoting  by  v  the  velocity  at  the  end  of  any  time  £,  we 


have, 


v2  = 


d  s1         dz2  -+■  dx2 


dt2 


dt2 


or,  replacing  the  values  of  dz  and  d x,  obtained  from  Equations  (160), 


v2  =  V2  -  2  V.g .  t.  sin  a  +  g2  fl 


(164) 


and   eliminating  t,    by    mears    of  the    first   of  Equations    (160),    and 
replacing   V2,  in   ths  last  term   by  its  value  2g  A, 


vi  —  v2  —  2g .  tan  a  .  x  -f-  g 


2  h    cos2  a 


.     •     • 


(165) 


134        elements   of   analytical  mechanics. 

in  which,  if  we   make   x  s=  4  h .  sin  a  cos  a,  we   have   the    velocity   at 
the   point  D, 

v2  =  F2, 

rthich  shows  that  the  velocity  at  the  furthest  extremity  of  the  range 
is   equal    to    the    initial    velocity. 

Differentiating  Equation  (161),  we  get 

dz  x 

—  z=  tan  6  =  tan  a  —  — —  ....     (166) 

ax  2  h.  cos2  a  v        ' 

in  which   &    is   the   angle  which   the   direction  of  the   motion   at   any 
instant   makes   with   the  axis   x. 
Making   tan  &  =  0,  we  find 

x  =  2  h  .  cos  a  .  sin  a, 
which,   in   Equation    (161),  gives 

z  =  h.  sin2  a, 

the   elevation  of  the   highest  point. 

Substituting  for  x,  the  range,  4  h  cos  a  sin  a,  in    Equation  (166), 

tan  6  —  —  tan  a, 

which    shows    that    the   angle  of  fall  is  equal  to  minus  the  angle  of 
projection. 

§  150. — The  initial  velocity  V  being  given,  let  it  be  required  to 
find  the  angle  of  projection  which  will  cause  the  trajectory  to  pass 
through   a   given  point  whose   coordinates  are  x  =  a  and   z  =  b. 

Substituting   these   in   Equation    (161),  we   have 


b  —  a  tan  a  — 


<72 


4  /* .  cos2  a 


from   which   to   determine   a. 
Making   tan  a  =  <p,  we   find 

1 

1   +  <Da 


MECHANMCS    OF    SOLIDS 


135 


ft-mcn    in   the    equation    above,    gives 

4h.b  -f  a2  —  4  A .  a .  9  +  a2  p2  =  0; 


whence, 


2/i        1     /TD T-T7 o 

<p  ==  tan  a  =  —  ±  -v/4r  -4Ao  -a2   • 

a         a 


(167) 


The  double  sign  shows  that  the  object  is  attained  by  two  angles, 
and  the  radical  shows  that  the  solution  of  the  problem  will  be 
possible   as   long   as 

4/i2  >4A6  +  a2. 

Making, 

4  k2  —  4  h  .b  —  a2  =  0, 

the  question  may  be  solved  with  only  a  single  angle  of  projection. 
But  the  above  equation  is  that  of  a  parabola  whose  co-ordinates  are 
a  and  6,  and  this  curve  being  con- 
structed and  revolved  about  its  vertical 
axis,  will  enclose  the  entire  space 
within  which  the  given  point  must  be 
situated  in  order  that  it  may  be  struck 
with  the  given  initial  velocity.  This 
parabola  will  pass  through  the  farthest 
extremity  of  the  maximum  range,  and 
at  a  height  above  the  piece  equal  to  k. 

§  151. — Thus  we  see   that   the  theory  of  the  motion  of  projectiles 
is  a  very  simple  matter  as  long  as  the  motion  takes  place  in   vacuo. 
But  in  practice  this  is  never  the  case,  and  where  the  velocity  is  con 
siderable,  the  atmospheric    resistance    changes  the  nature  of  the   tra- 
jectory, and  gives  to  the  subject  no  little  complexity. 

Denote,  as  before,  the  velocity  of  the  projectile  when  the  atmos- 
pheric resistance  equals  its  weight,  by  k,  and  assuming  that  the 
resistance  varies  as  the  square  of  the  velocity,  the  actual  resistance 
at  any  instant  when  the  velocity   is  v,  will  be, 


M .  g  .  v2 


Mcv2, 


136  ELEMENTS    OK     ANALYTICAL    MECHANICS. 

by  making, 

*-• 

The  forces  acting  apon  the  projectile  after  it  leaves  the  piece 
being  its  weight  and  the  atmospheric  resistance,  Equations  (120), 
become, 

d2x 
M •  -r-z-  =  M  g .  cos  a  -f  Mc.v2 .  cos  a', 
a  t2 

M.tl.  =  Mg.  cos  0  +  Mc.v2.  cos  /3', 

a  r 

d2  z 
M'  •—•  —  Mg .  cos  y  +  Mc .  v2 .  cos  y* 

(Jb     L 

Taking  the  coordinates   z   vertical,    and    positive    when   estimated 
upwards, 

cos  a  =  0 ;     cos  jS  =  0 ;     cos  y  =  —  1, 

and  because  the  resistance  takes  place  in  the  direction  of  the  trajec- 
tory, and  ill  opposition  to  the  motion,  if  the  projectile  be  thrown  in 
the  first  angle,   the  angles  a',  /3',  and  y\    will  be  obtuse, 

dx  nt  dy  dz 

cos  a'  = ; —  :     cos  p'  = r—  ;     cos  y'  = 7— , 

d  s  as  «« 

and  the  equations  of  motion  become,  after  omitting  the  common 
factor  My 

d2  x  0     d  x 

— — -  =  —  c  •  vl  •  — : —  : 
dt2  ds  ' 

&y  o   dv 

— —  =•  —  c  •  vz •  — — -  : 
dt2  ds  ' 

d2  z  ,     dz 

dt2  y  ds 

From  the  first  two  we  have,  by  division, 

d2y         d2x 
dy         dx  * 


MECHANICS    OF    SOLIDS.  137 

and   by  integration, 

log  dy  =  log  dx  -f-  log  (7; 

and,  passing  to  the  quantities, 

dy  =  Cdx, 

Integrating  again,  we  have, 

y  =   Cx+   C; 

in  which,  if  the  projectile  be  thrown  from  the  origin,  C  =  0,  .hus 
giving  an  equation  of  a  right  line  through  the  origin.  Whence  we 
see  that  the  trajectory  is  a  plane  curve,  and  that  its  plane  is  vertical 
through  the  point  of  departure. 

Assuming    the   plane   z  x,  to    coincide   with   that  of  the   trajectory, 
and  replacing  v2,  by  its  value  from  the  relation, 


we  have, 

<Px 
.      ~dt2  ~~ 

d2z 
~JT2  '" 

From  the  first  we  have, 

d2x 


dt2  ~~ 

V2, 

—  c  - 

ds 
dt 

d  x 
dt 

• 

t 

-  9 

—   €' 

ds 

dz 

• 

j  * 

(168) 


dt2  ds 


d  x  dt 


dt 


and   by  integration, 


los   4r=  -c-s+°- 


Denoting  by  c,  the   base  of  the   Naperian   system   of   logarithm*, 
and  making  C  =  log  A,  the  above  may  be  written, 

d  x 

1°8    ~s~  —  —  c '  8  X  1°#  e  ■#■  loar  A, 

at  9 


138  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and  passing  from   logarithms  to  the  quantities, 

d  x  —  c  * 

in  =  A-f     m 

Denoting  by    V,  the  initial    velocity,  and   b)    a,  the   angle  of  pro- 
jection, we  have,  by  making  s  =  0, 

d  x 

— - —  =  A  =  V  cos  a, 

dt  ' 

which   substituted   above,  gives 

dr  ~c' 

—   =  F.cosa.e  (170) 

To    integrate   the  second    of  Equations   (168),  make 

dz  dx  /«**.\ 

—  =».—, (171) 

dt         r    dt  V         ' 

in   which  p  is   an  additional    unknown    quantity. 

Differentiating    this    equation,     dividing    by    dt,    and    eliminating 

d2  x 
from    the   result,  -r-y*    by    its    value   in    the   first  of  equations  (168), 

we   have, 


<Pz  dp     dx  ds   dx 

d~fi         dt'  d~t'~  P'C'  d~t"dt 

and   substituting   this   value   in    the    second   of  Equations    (168),    we 

d  z 
have,  after  eliminating    —  by  its  value,  obtained  from  Equation  (171), 

£.*:***, (172) 

dt     dt  y  v       ; 

and    dividing   this  by   the  square   of  Equation   (170), 

dp 

d  t  g  2et 

dx~~    V2  cos2  a  '6  '178^ 

di 


MECHANICS    OF    SOLIDS.  139 

but   regarding  z   and  p  as  functions  of  x,  we   have,  Equation  (171), 

dJ. 

*  =  5=  -dir <174> 

dt 

and, 


dp 

dt 

dp 

dx 

dx 

dt 
whence,  making    V2  =  2g  A,  Equation   (173)  becomes 


,  let 

dp  e 

-i    »     •     •     •     • 


dx  2 h .  cos2  a 

and  multiplying  this  by  the  identical  equation, 


(175) 


d  x  .  -/I  -f  p2  =  ds, 
obtained  from  Equation  (174),  we  find, 


2ct 

e  •  ds 


and    integrating, 


let 
e 


in  which  C  is  the  constant  of  integration  ;   to  determine  which,  make 
8  =  0  ;  this  gives  p  =z  tan  a ;  and 

1  . . 

C  =  — jr-—  +  tan  a .  y  1  -f  tan2  a  +  log  (tana  Tyi+  tan2a)  •  (177) 

From    Equation  (175)  wre   have, 

—  2  c  * 
dx  =  —  2  A.  cos2  a  .  e        •  rfp 

from   Equation   (171), 

dz  z=  p  .dx  \ 


140         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

from  Equation   (172), 

g  d  fi  =  —  d  x .  dp  ; 

and   eliminating   the  exponential  factor  by  means  of  Equation   (176), 
we  find. 

c.dx  = £ ;   .    (178) 

p  «/ 1  +  Pl  +  log  (p'+  ■/!  +  f)  -  C 

c.dz  = - P^ - ;  •     (179) 


p  V {  +  p2  +  ]°g  (p  +  V l  +  p2)  - 

i 

■  c 

pd  p 

p  V1  +  p2  +  ]°g  (p  +  V1  +  p2)  - 

-  c 

—  dp 

V~^9  ,dt=  ■  {  ;  •   (180) 

J  V  -P  V1  +  P2  ~  loS  ( P  +  V  J  +  ^2) 

Of  the  double  sign  due  to  the  radical  of  the  last  equation,  the 
negative  is  taken  because  p,  which  is  the  tangent  of  the  angle  made 
by  any  element  of  the  curve  with  the  axis  of  x,  is  a  decreasing 
function  of  the  time  t. 

These  equations  cannot  be  integrated  under  a  finite  form.  But 
the  trajectory  may  be  constructed  by  means  of  auxiliary  curves  of 
which  (178)  and  (179)  are  the  differential  equations.  From  the  first. 
we  have, 

dx  =  T  .dp; (181) 


and  from  the  second, 


in  which, 


dz  =  T.p.dp; .(182) 


T  = l ;  •  (183) 

c      p  y/T~+~P2  +  log  (p  +  V^  +P2)  ~  0 

and  dividing  Equations  (181)  and    (182),  by  dp, 

d  x 


dp 

dz 
dp 


T;    .     . (184) 


=  r.i>; (185) 


MECHANICS    OF    SOLIDS. 


143 


r 

(1.) 

m 

A 

9                  A 

c                       2 

Now,  regarding  x,  p,  and  z,  p,  as  the  variable  co-ordinates  of  two 
auxiliary  curves,  T,  and  T .  p,  will  be  the  tangents  of  the  angles 
which  the  elements  of  these  curves  make  with  the  axis  of  p. 

Any  assumed  value  of  p,  being  substituted  in  T,  Equation  (183), 
will  give  the  tangent  of  this  angle,  and  this,  Equation  (184),  multi- 
plied by  dp>,  will  give  the  difference  of  distances  of  the  ends  of  the 
corresponding  element  of  the  curve  from  the  axis  of  p.  Beginning 
therefore,  at  the  point  in  which  the  auxiliary  curves  cut  the  axis  of 
p,  and  adding  these  successive  differences  together,  a  series  of  ordi- 
nates  x  and  z,  separated  by  intervals  equal  to  dp,  may  be  found,  and 
the  curves  traced  through  their   extremities. 

At  the  point  from        — ^  E 
which    the     projectile 
is  thrown,  we  have, 

z=0]  z  =  0  ;  jo=tana, 

and  the  auxiliary 
curves  will  cut  the 
axis  of  p,  in  the  same 
point,  and  at  a  dis- 
tance from  the  origin  equal  to  tan  a.  Let  A  B,  be  the  axis  of  p, 
and  A  C,  the  axis  of  x  and  of  z ;  take  A  B  =  tan  a,  and  let  BzD, 
and  BxE,  be  constructed  as  above. 

Draw  the  axes  Ax  and  Az,  through  the  point  of  departure  A, 
Fig.  (2)  ;  draw  any 
ordinate  c  zt  xd  to  the 
auxiliary  curves  Fig. 
(1);  lay  off  ^Fig. 
(2)  equal  to  Cxt  Fig. 
(1),  and  draw  through 
xt ,  the  line  xt  zt 
parallel  to  the  axis 
Az,  and  equal  to  czt 
Fig.  (1)  ;  the  point 
zt  will    be  a  point    of 

the  trajectory.      The  range   A  D,  is  equal  to  ED,  Fig.  (\\. 

10 


142  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

By  reference  to  the  value  of  (7,  Equation  (177),  it  will  be  seen 
that  the  value  of  Tr  Equation  (183),  will  always  be  negative,  and 
that  the  auxiliary  curve  whose  ordinates  give  the  values  of  x,  can, 
therefore,  never  approach  the  axis  of  p.  As  long  as  p  is  positive, 
the  auxiliary  curve  whose  ordinates  are  z,  wrill  recede  from  the 
axis  p ;  but  when  p  becomes  negative,  as  it  will  to  the  left  of 
the  axis  A  C,  Fig.  (1),  the  tangent  of  the  angle  which  the  element 
of  the  curve  makes  with  the  axis  p,  will,  Equation  (185),  become 
positive,  and  this  curve  will  approach  the  axis  p,  and  intersect  it  at 
some   point  as  D. 

The  value  of  p  will  continue  to  increase  indefinitely  to  the  left 
of  the  origin  A,  Fig.  (1),  and  when  it  becomes  exceedingly  great^ 
the  logarithmic  term  as  well  as  67,  and  unity  may  be  neglected  in 
comparison  with  p,  which  will  reduce  Equations  (178)   and  (179)  to 

c  .p 

z  —  C"  -\ •  log  », 

c p '  c 

which  will  become,  on  making  p  very  great,. 

x  =  C";     z  =  C"  +  -log  jo, 

c 

which  shows  that  the  curve  whose  ordinates  are  the  values  of  a*, 
will  ultimately  become  parallel  to  the  axis  p,  while  the  other  has 
no  limit  to  its  retrocession  from  this  axis.  Whence  we  conclude, 
that  the  descending  branch  of  the  trajectory  approaches  more  and 
more  to  a  vertical  direction,  which  it  ultimately  attains ;  and  that 
a  line  G  L,  Fig.  (2),  perpendicular  to  the  axis  x,  and  at  a  distance 
from  the  point  of  departure  equal  to  C,  will  be  an  asymptote  to 
the  trajectory. 

This  curve  is  not,  like  the  parabolic  trajectory,  symmetrical  in 
reference  to  a  vertical  through  the  highest  point  of  the  curve ; 
the  angles  of  falling  will  exceed  the  corresponding  angles  of  rising, 
the  range  will  be  less  than  double  the  abscissa  of  the  highest  point, 
and  the  angle  which  gives   the  greatest  range  will   be  less  than  45°. 


' 

d  x 

dp 
c  .p2 

and   integrating, 

x  = 

=  C  - 

1 

MECHANICS    OF     SOLIDS. 


143 


Denoting    the   velocity   at   any   instant  by    v,   we   have 


p 


dx2  -f  dz2  dx* 

—  =  (i  +  jp2)  ■ 


r/*2 


rf*2 


and  replacing    c?x2    and    d*2   by  their  values    in  Equations  (178)  and 
(180).  we  find 

1  9.{l+p2) 


v 


2    _ 


C        C  -p   y/\    +  p2   -  log  (p  +    -/l    +  J92) 


(180) 


and  supposing  />  to  attain  its  greatest  value,  which  supposes  the 
projectile  to  be  moving  on  the  vertical  portion  of  the  trajectory, 
this  equation  reduces,  for  the   reasons   before  stated,  to 


•  =  v/t=^ 


which  shows    that   the   final   motion  is  uniform,  and  that  the  velocity 
will    then    be   the    same   as    that   of  a   heavy    body    which  has  fallen 

1  k2 


in   vacuo  through  a  vertical  distance  equal  to 


2c 


2<7 


§152. — When  the  angle  of  projection  is  very  small,  the  projectile 
rises  but  a  short  distance  above  the  line  of  the  range,  and  the  equation 
of  so  much  of  the  trajec- 


Z 


tory  as  lies  in  the  imme- 
diate neighborhood  of  this 
line  may  easily  be  found. 
For,  the  angle  of  projec- 
tion being  very  small,  p 
will  be  small,  and  its 
second  power  may  be 
neglected  in  comparison 
with    unity,    and   we  may 

take, 

d  s  =  d  x ;    and    *  =  x ; 

which  in  Equation  (175),  gives, 

2c* 


3* 


J) 


dp 
~dx~ 


d2z 
dx1 


2  h  .  cos2  a 


(187; 


144         ELEMENTS    OF    ANALYTICAL    MECHANICS. 
Integrating, 

2  ex 

dZ  '  +0: 


dx  \cji.  cos2  a 

dz 
makirg  x  =  0,  we  have  — —  =  tan  a, 

whence, 

C  =  tan  a  + 


4  c  .  h  .  cos2  a 
which  substituted  above,  gives, 

2c* 

e?z  e 

=  tan  a  — - — \- 


dx  4  c  .  h  .  cos2  a       4  c  .  h  .  cos1  a  ' 

«nd  integrating  again 

lex 
€  X 

z  =  tan  a .  s  —  — — —  +  - — -—  4.  C\ 

8c2  .  h  .  cos2  a      \c.h.  cosz  a 

making  #  =  0,  then  will  z  =  0,  and 

1 

(7'  = 


8  c2 .  /i .  cos2  a 
hence, 

z  -.  tanaar—  — — ~-  (e  *   —  2tx  —  l)  .     .     (188) 

8  c2 .  A  .  cos2  a    V  / 

From  Equation  (172),  we  have, 

g  .dt2  =  —  dx  .  dp, 
and  substituting  the  value  of  dp,  from  Equation  (187), 

ex 

-  e      .  dx 

a  t  = 


-y/2  <jr  A  .  cos  a ' 
Mid  integrating,  making  ar  =  0,  when  t  =  0, 

.    (/*«  1)     .     .     .     .      (189) 


*  = 


C  1/  "2  q  h  .  cos  a 


MECHANICS    OF    SOLIDS.  145 

which   will   give   the   time   of    flight    to   any    point   whose   horizontal 
distance  from    the   piece   is   equal   to   z. 

§  153. — Let  the  projectile  fall  to  the  ground  at  the  point  Z),  and 
denote  the  co-ordinates  of  this  point  by  x  =  /,  and  z  =  X,  and  sup- 
pose the  time  of  flight  or  t  =  <r.  These  values  in  Equations  (188) 
and    (189),  give 

—  8c2. h.  cos2a  (X  -  /.tana)  =  ecl  —  2cl  —  1    •    (189)' 

/ c  l 

cos  a.<r.c.y2gh  =  e      —1      •     •     •    '«  (189)" 

When  the  two  constants  h  and  c,  as  well  as  a  and  X,  are  known, 
these  equations  will  give  the  horizontal  distance  /,  and  the  time  of 
flight.  Conversely,  when  the  quantities  a,  /,  X  and  r  are  known, 
they  give  the  co-efficient  of  resistance  c,  and  the  height  A,  due  to 
the  velocity  of  projection,  and  therefore,  Equation  (135),  the  initial 
velocity  itself. 

Eliminating  the  height  A,  we  find 

-  4  (X  -  l.tma)(ecl  -  l)2  =  g.<r*.  (e**1  -  2cl  -  1)  ;  .  (189)'" 

from  which   the  value  of  c  may  be  found,  and    one  of  the  preceding 
equations  will   give  A,  or   the  initial  velocity. 

It  may  be  worth  while  to  remark  that  if  the  exponential  term 
in  Equation  (188)  be  developed,  and  c  be  made  equal  to  zero,  which 
is  equivalent  to  supposing  the  projectile  in  vacuo,  we  obtain  Equa- 
tion (161). 

§  154. — Assuming  that  the  resistance  of  the  air  varies  as  the  squaie 
of  the  velocity,  some  idea  may  be  formed  of  its  actual  intensity  from 
the  fact  that  a  twenty-four-pound  ball  projected  with  a  velocity  of  2,000 
feet  ill  vacuo,  and  under  an  angle  of  45°,  would  have  a  range  of 
125,000  feet;  whereas  actual  experiment  in  the  air  shows  it  to  be  but 
7,300  feet — about  one-seventeenth  of  the  former. 

Many  circumstances  qualify  both  the  path  and  velocity  of  projectiles. 
The  law  of  the  resistance  may  be  the  same  for  all  figures,  but  it  is 
known,  from  actual  trial,  not  to  be  that  of  the  square  of  the  velocity, 
except  for  very  small  rates  of  motion.      For  the  same  velocity,  the  in 


14:6  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

tensity  of  the  resistance  varies  with  the  size  and  figure  of  the  ball.- 
Much  depends  upon  the  facility  with  which  the  compressed  air  in  front 
may  escape  latterly  and  make  its  way  to  the  rear.  The  actual  resist 
ance  at  any  instant  is  composed  of  two  terms,  the  one  due  to  the 
inertia  of  the  displaced  particles,  the  other  to  the  diffcrenc  of  at- 
mospheric pressure,  as  such,  in  front  and  rear.  If  during  the  motion 
the  air  could  close  in  behind  and  exert  the  same  pressure  as  in  front, 
the  resistance  would  be  wholly  due  to  inertia.  If  the  ball  were  at  rest, 
and  all  the  air  removed  in  rear  of  the  plane  of  largest  section  perpen- 
dicular to  the  trajectory,  the  resistance  would  be  due  entirely  to  the 
barometric  pressure  on  the  extent  of  this  section.  Both  terms  of  the 
resistance  must  be  variable  and  a  function  of  the  velocity,  till  the  latter 
is  so  great  as  to  leave  a  vacuum  behind,  when  the  barometric  term 
would  become  constant. 

From  a  careful  and  elaborate  investigation  of  the  numerous  experi 
ments  upon  this  subject,  Col.  Piobert  has  constructed  this  empirical 
formula  for  spherical  projectiles,  viz. : 

p=A.7T.r'(l  +  V-yv> (       ) 

in  which  p  is  the  resistance  in  kilogrammes,  v  the  velocity,  n  the  ratio 
of  the  diameter  to  the  circumference,  r  the  radius  of  the  ball,  A  the 
resistance  on  a  square  mP.tre  when  the  velocity  is  one  t»  etre,  and  9  the 
velocity  which  wrould  make  the  resistance  measured  by  the  second  term 
equal  to  that  measured  by  the  first. 

§  155. — If  the    ball    be    not    perfectly  homogeneous   in    density,  the 

centre  of  inertia  will,  in  general,  be  removed   from  that  of  figure;  the 

resultant  of  the  expansive  action  of  the  powder  will  pass  through  the 

latter  centre   and   communicate  to  the  ball  a  rotarv  motion    about  the 

former.  The  atmospheric  resistance  will  be  greater  on  the  side  of  the 
greatest  velocity,  and  deflect  the  projectile  to  the  opposite  side. 


MECHANICS    OF    SOLIDS. 


147 


EOTAKY    MOTION. 

§  156. — Having  discussed  the  motion  of  translation  of  a  single 
body,  we  now  come  to  its  motion  of  rotation.  To  find  the  circum- 
stances of  a  body's  rotary  motion,  it  will  be  convenient  to  transform 
Equations  (118)  from  rectangular  to  polar  co-ordinates.  But  before 
doing  this,  let  us  premise  that  the  angular  velocity  of  a  body  is  the 
rate  of  its  rotation  about  a  centre.  The  angular  velocity  is  measured 
by  the  absolute  velocity  of  a  point  at  the  uniCs  distance  from 
the  centre,  and  taken  in  such  position  as  to  make  that  velocity  a 
maximum. 


§  157. — Both    members   of  Equations    (38)    being  divided   by  d  t. 


give 


dz' 

~~dt 

djf_ 
dt 

dz' 
dt 


=  z 


=    X 


=  y 


d-^ 

IT 

d  9 
~~dl 

d-ti 
~d7 


-  r 


—  z 


—    X 


d  <p 
'~dJ 

d-d 

~dT 

d^ 
dt 


(190) 


in  which  the  first  members  taken  in  order,  are  the  velocities  of  any 
element,  as  m.  in  the  direction  of  the  axes  x,  y,  z,  respectively,  in 
reference  to  the  centre  of  inertia,  §  75,  while 


dix       d\       dtp 


dt 


; — J 


dt       dt 


are  the   angular  velocities   about   the   same  axes  respectively. 

Denoting  the  first  of  these  by  v,,  the  second  by  vy,  and  the  third 
by  vf,  we  have 


dts 
H 


d\ 


rf<p 


~  V-'     dt      '  V     dt    m  '•' 


(191) 


148 


ELEMENTS    OF    ANALYTICAL    MECHANICS 


and  Equations  (190)  may  be  written 

dx' 

_=*.v,-y.v„ 

dy' 


dt 


=  z'.v,  ~  *\*., 


(192) 


§  158. — If  an  element  m  be  so  situated  that  its  velocity  shall  be 
equal  and  parallel  to  that  of  the  centre  of  inertia,  then,  for  this 
element,  will  each  of  the  first  members  of  Equations  (192)  reduce 
to   zero,  and 

x'  .vx  —  z'  ,vx  =  0, 


y' .  v,  —  x' .  vy  =r  0  ; 


V 


(193) 


the  last  being  but  a  consequence  of  the  two  others,  these  equation* 
are  those  of  a  right  line  passing  through  the  centre  of  inertia, 
every  point  of  which  will  have  a  simple  motion  of  translation 
parallel  and  equal  to  that  of  the  centre  of  inertia.  The  whole 
body  must,  for  the  instant,  rotate  about  this  line,  and  it  is,  there- 
fore, called    the  Axis  of  Instantaneous  Rotation. 

§159. — Denote  by  ay , 
ft  j  yt ,  the  angles  which 
this  axis  makes  with  the 
co  ordinate  axes  a:,  y,  2, 
respectively.  Then,  tak- 
ing any  point  on  the  in- 
stantaneous axis,  will, 


cos  a;  = 


cos/3,  = 


VV2  +  y">  -f-  g* 

V 

-vA'2  +  y'2  +  z** 


>  < 


cos/,  = 


V*'2  +  y'*  -r  z'2 


MECHANICS     OF     SOLIDS. 


m 


and  eliminating  .r',  y'  and  z\  by  Equations  (193), 


cos  at  = 


VV  +  V  +  v.2 


cos/3,  = 


VV  +  \2  +  v22 


:>  > 


cos  7,  = 


VV  +  %*  +  Vz2 


(194) 


which  will  give    the    position    of    the   instantaneous   axis    as   soon  as 
the  angular  velocities  about  the  axes  are   known. 

§160. — Squaring    each   of   Equations    (192),  taking   their   sum   and 
extracting   square  root,  we  find 


j 


dx'2  +  dy'2  -+-  dz'2 


dt2 


=v  =  S(z\vy-y\y  +  (x\v-z\vz)H(/-\-*'.vy)2', 


Replacing  vz ,  vy  and  v2  by  their  values  obtained  by  simply  clearing 
the   fractions  in   Equations    (194),    this   becomes 


v=  yvz2  H~  vy2  -f  v22  x  yV2  +  y"1  -f  z'2  —  (x'  cos  a,  -f  y'cos/3,  +  z'  cos  y,)2, 

which  is  the  velocity  of  any    element   in    reference   to  the  centre  of 
inertia. 
Making 

x'2  -f  y'2  +  z'2  =  1, 
we  have  the  element  at  a  unit's  distance  from  the  centre  of  inertia; 


and  making 


x'  cos  at  -f  yr  cos  fit  -\-  z'  cos  yt  =  0, 


(195) 


the  point  takes  the  position,  giving  the  maximum  velocity.  In  this 
case  v  becomes  the  angular  velocity,  and  we  have,  denoting  the 
latter  by  v|f 


h  =  yV  +  v  2  +  v,»  •     • 


(196) 


150  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Equation  (195)  is  that  of  a  plane  passing  through  the  centre 
of  inertia,  and  perpendicular  to  the  instantaneous  axis.  The  position 
of  the  co-ordinate  axes  being  arbitrary,  Equation  (196)  shows  that 
tb<i  sum  of  the  squares  of  the  angular  velocities  about  the  three 
coordinate  axes  is  a  constant  quantity,  and  equal  to  the  square  of 
the  angular  velocity  about   the   instantaneous   axis. 

§  161.— Multiply  Equation  (196),  by  the  first  of  Equations  (194), 
aud  there  will  result 

Vj.  .  cos  a4  =  »z     •'•••*•••    (197) 

whence  the  angular  velocity  about  any  axis  oblique  to  the  instanta- 
neous axis,  is  equal  to  the  angular  velocity  of  the  body  multiplied 
by  the  cosine  of  the  inclination  of  the  two  axes. 

§162. — Equation  (196)  gives  vi9  when    vx,v  ,  vz,  are  known.     To 

find  these,  resume  Equations  (118),  and  write  for  the  moments  of  the 
extraneous  forces  in  reference  to  the  axes  #,'  y[  z,'  through  the  centre 
of  inertia,  Nn  Mt,  L(,  respectively,  then  will 


/d2y'  ,  d2x'  A 

/d2x'  ,  d2z'  A 

\dt2  d  t2  /  ' 

(d2z'  ,  d2y'  ,\ 


y  .     •     •     (198) 


j 


dt2     '  dt2 

differentiating  the  first  of  Equations  (192),  with  respect   to  t,  we  find 
cPxr  dz'  dy'         dv  dv 

—  y     • V     •  — —    -4-    y  •  Z    — •  V   5 

dt2  y     dt  l     dt  dt  dt      u 

and  replacing  — —  and  —~  >  by    their    values    given    in    the    second 
^        b    d  t  dtJ  fo 

and  third  of  Equations   (192), 

<Px'  /   „    .      A       ,   .  ,   .  .    .    dv„     ,       •*, 


dt2 


=  -  (v  +  '*2)  -x'  +  vv#  +  v*-v* • «'  +  it'z'  ~ Tt'y' ; 


MECHANICS    OF    SOLIDS. 


151 


in  the  same  way 


and  these  values  in  the  first  of  Equations,  (198),  give 


/(Py'     ,     d2x'     \ 


+vy.vx.2m.  (a?'2-/2) 

Similar  equations  will  result  from  the  remaining  two  of  Equations 
(198)  ;  then  by  elimination  and  integration,  we  might  proceed  to  find 
the    values    of  vz ,    v     and    vz ,    but    the    process    would   be   long   and 

tedious.  It  will  be  gr<*itly  simplified,  however,  if  the  co-ordinate 
axes  be  so  chosen  as  to  make  at  the  instant  corresponding  to  f, 


2  m  x'  y'  =  0  ;    2  m  z'  y'  =  0 ;    2  m  z'  x'  =  0  ; 


(200) 


which    is    always   possible,    as    will    be    shown    presently.      This  will 
reduce  Equation  (199)  to 

.  ^.i^f  +  ^  +  ^.^.^^-y^v 

The   other    two    equations   which    refer   to   the   motion    about    the 
axes  y'  and  x\  may  be  written  from    this   one.     They  are, 

^>  .  2  m  {**  +  z'2)  -»-  vz  .  vz .  2  m  (aft  -  *'2)  =  Mt , 

~"  .2  m  (y'2  +  z"1)  -f  vy  .  vz  .  2  m  (v'»    -  z'%)  =  X> . 


152  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

The  axes  *',  y\  z\  which  satisfy  the  conditions  expressed  ic 
Equations  (200),  are  called  the  principal  axes  of  figure  of  the  body. 
And  if  we   make 


2  m  .  (y'2  +  x'2)  =   09   } 
2  m  .  (x'2  +  z'2)  =  B, 
2  m 


.  {x'2  +  z'2)  =  B,     „ 
.  (y'a  +  z'2)  =  A;J 


we  find,  by  subtracting    the    third    from   the  second, 

2  m  .  {x2  -  y'2)  =  B  -  A, 
the    first   from    the    third, 

2»,  (*'2  -  x'2)  =  A  -  C, 
and   the   second   from  the    first, 

2  m  .  (y'2  -  z'2)  =   C  -  B\ 


(201) 


which   substituted   above,  give, 


d 


*.£»  +  .,.vU- 


A)-. 

=  A? 

C)  = 

*M„ 

Jt)  = 

:*.- 

(202) 


By  means  of  these  equations,  the  angular  velocities  vz ,  v     vt ,  must 
be  found  by  the   operations  of  elimination   and  integration. 

It  is  plain  that  the  quantities  C,  B  and  A,  are  constant 
for  the  same  body  ;  the  first  being  the  sum  of  the  products  arising 
from  multiplying  each  elementary  mass  into  the  square  of  its  dis- 
tance from  the  principal  axis  z\  the  second  the  same  for  the  prin- 
cipal axis  y',  and  the  third  for  the  principal  axis  x'.  The  sum 
of  the  products  of  the  elementary  masses  into  the  square  of  their 
distances  from    any  axis,  is  called   the  moment  of  inertia  of  the  body 


MECHANICS    OF    SOLIDS. 


153 


in  reference  to  this  axis,  and  it  measures  the  capacity  of  the  body  to 
store  up  work  in  the  shape  of  living  force  during  a  motion  of  rotation 
about  that  axis.     A,  B,  and   C  are  called  principal  moments  of  inertia. 

§  163. — Through  any  assumed  point  there  may  always  be  drawn 
one  set  of  rectangular  axes,  and,  in  general,  only  one  which  will  satisfy 
the  conditions  of  Equations  (200).  To  show  this,  assume  the  formulas 
for  the  transformation  from  one  system  of  rectangular  axes  to  another, 
also  rectangular.     These  are 

x  =  x .  cos  {x'  x)  -f-  y .  cos  {x'  y)  -f-  z  .  cos  {x'  z),  y 

y'  =  x.co%(yx)  +  y.cos(y'y)  -f  z  .  cos  (y' z),   I  .     .     (203) 

z'  =  x .  cos  (zr  x)  -f-  y .  cos  (z'  y)  -j-  z .  cos  (z'  z), 

in  which  (-r'#),  (y'#),  and  (z' #),  denote  the  angles  which  the  new 
axes  x,  y\  z'y  make  with  the  primitive  axis  of  x\  (x'  y)y  (y' y),  and 
(z'  y),  the  angles  which  the  same  axes  make  with  the  primitive  axis 
of  y,  and  (x'  z),  (y'  z),  and  (z'  z),  the  angles  which  they  make  with 
the  axis  z. 

Assume  the  common 
origin  as  the  centre  of  a 
sphere  of  which  the  radius 
is  unity ;  and  conceive  the 
points  in  which  the  two 
sets  of  axes  pierce  its  sur- 
face to  be  joined  by  the 
arcs  of  great  circles;  also 
let  these  points  be  con- 
nected with  the  point  JVy 
in  which   the    intersection 

of  the  planes  xy  and  x'y'  pierces  the  spherical  surface  nearest  to  that 
in  which  the  positive  axis  x  pierces  the  same.     Also,  let 
0  =  Z'AZ  =z  X'JYX,  being   the  inclination  of  the  plane  x'y'  to   that 

of  xy. 
tp  =  NAX,   being   the   angular   distance   of    the    intersection    of    the 

planes  xy  and  x'y'  from  the  axis  x. 
<f>  =  NAX,  being  the  angular  distance  of  the  same  intersection  from 

the  axis  x ', 


154  ELEMENTS     OF    ANALYTICAL    MECHANICS 

Then,  in    the   spherical    triangle  X  NX, 

cos  (x1  x)  =  cos  4 .  cos  9  -f-  sin  y  .  sin  (p .  cos  6  ; 

In  the  triangle  Y'  NX,  the  side  N  Y'  —  -5-  +  <p.  and 

cos  (yr  x)  =  —  cos  4>  •  sin  9  4-  sin  -^ .  cos  9  .  cos  B9 

In    the   triangle  Z'  NX,  the  side  NZf  =  —•>  and 

SB 

cos  (z'  a;)  =  sin  -^ .  sin  0. 

And   in   the   same   way  it  will   be  found  that 

cos  (#'  y)  =  —  sin  4*  •  cos  9  -f-  cos  4 .  sin  9 .  cos  I  j 
cos  (y'  y)  =  sin  4  .  sin  9  -f-  cos  %}/ .  cos  9 .  cos  6  ; 
cos  (z'  y)  =  cos  4*  •  sin  4  ; 
cos  (xr  z)  =  —  sin  9  .  sin  6  ; 
cos  (yf  z)  =.  —  cos  9  .  sin  4  ; 
cos  (0'  z)  =  cos  6  ; 

and   by  substitution   in   Equations  (203), 

x'  =  x  (sin  4 .  sin  9 .  cos  &  -}-  cos  4 .  cos  9) 

-f-  y  (cos  -^  •  sin  9 .  cos  &  —  sin  -^ .  cos  9)  —  z  sin  9 .  sin  6 
yf  =  x  (sin  4  •  cos  9 .  cos  &  —  cos  4  •  sin  9) 

4*  y  (cos  4  •  cos  9  .  cos  &  -f-  sin  4  •  sin  9)  —  z  cos  9  .  sin  6% 
zr  as  #  sin  -^  •  sin  0  4-  y  cos  4>  •  sin  0  -f-  0  cos  6  ; 

or   making,  for  sake  of  abbreviation, 

D  =  x  cos  -^  —  V  sin  4, 

j£  =  x  sin  -^  .  cos  &  +  y  cos  4/ .  cos  6  —  2  sin  d, 

the  above    reduce   to 

x'  =  E .  sin  9  +  &  •  cos  9, 
yf  =  .#.  cos  9  —  D  .  sin  9, 
f '  =  a: .  sin  4 .  sin  ^  +•  y  .  cos  4  .  sin  6  -f  2 .  cos  6, 


MECHANICS     OF    SOLIDS.  155 

Substituting   these  values  in   the   equations 

lm.xf  .1/  =  0;     2  m .  ** .  z'  =  0  ;     2w.y',2'  =  0; 

we  obtain  from  the  first, 

sin  9  .  cos 9 . 2 m  (E2  —  D2)  +  (cos2 9  —  sin2 9)  ^mE.D  =  0, 

or,  replacing  sin  <p.cos<p,  and  cos2  9  —  sin2 9,  by  their  equals  |  sin  2  9, 
and  cos  2 9,  respectively, 

sin  29.  2  m(^2  -  I)2)  +  2  cos  29.2  mi>.  ^  =  0;  ■  •  •  (204) 
and  from    the    third    and   second,  respectively, 

cos  9  .  2  m  .  E .  z'  —  sin  9  .  2  m  D  .  z'  =  0,  •     •     •     (205) 
sin  9  .  2  m  .  E.  z'  +  cos  9  .  2  m  D  .  z'  =  0.  •     •     •     (206) 

Squaring  the  last  two  and  adding,  we  find 

(2mJ.z')2  +  pffl.i).2')2  =  0. 
which  can  only  be  satisfied  by  making 

s«.z>.,'  =  o.( •  <207) 

These  equations  are  independent  of  the  angle  9,  and  will  give  the 
values  of  4*  and  & ;  and  these  being  known,  Equation  (204)  will  give 
the  angle  9. 

Replacing  E  and  D  by  their  values,  we   have 

E .  z'  =  sin  d  .  cos  &  (x2  sin2  -^  4-  2  x  y  sin  \  cos  \  -\-  y2  cos2  4>  —  z2) 
4-  (cos2  &  —  sin2  &)  (x  z  sin  4>  +  y  z  .  cos  4>) , 

i) .  2'  =  sin  6  \xy  (cos2  4>  —  sin2  40  +  (x2  —  y2)  sin  4>  cos  4'} 
-|-  cos  &  (x  z  cos  4>  —  y  2  sin  40  • 

and  assuming 

2  w  a:2   =  ^4' ;  2my2  =  5';  2  m  z2  =  C ; 
2mxy  =  E';  2mxz  =  F' \  2myz  —  H\ 

and  replacing  sin  6  .  cos  0,  and  cos2  0  —  sin2  6,  by  their  respective 
values,  \  sin  2  0,  and  cos  2  0,  Equations  (20V)  become 


sin  2d  {A'  sin2 4,  +  2 ^' sin  +  cos  +  +  £'cos24,  -  C) 
+  2cos2d(/vsin4,  +  #'cos40 


1=., 


156  ELEMENTS    OF    ANALYTICAL    MECHANICS. 


sin  6  \E' .  (cos2  +  -  sin2  4)  +  (A'  -  B')  .  sin  4,  cos  +} 


4-  cos  d  (Ff  cos  4/  —  H'  sin  >L) 


f- 


in   which    A',  B\   C,  E\  F'  and   H\  are   constants,    depending   only 
upon  the    shape  of  the    body  and    the    position  of  the  assumed    axes* 

*i  y,  *> 

Dividing    the    first    by    cos  2  0,  and   the    second    by    cos  0,   they 
become 


tan  2  6  .  (-4'  sin2  4,  +  2  JF  sin  +  cos  4,  -f  Jff  cos  2 

4-  2  (i^  sin  4,  4-  H'  cos  4,) 


f        C'H=0;(20V)' 

tan  4  .  \E'  (cos2  4,  -  sin2  4,)  +  (v4'  -  £')  sin  \  cos  >L|  ) 

4-  i^cos^  -  #'sin4  J=-(20) 

From  the  first  of  these  we  may    find  tan   2  &,  and   from  the    second, 
tan  0,  in  terms  of  sin  4',  and  cos  4* ;  and  these  values  in  the  equation 

2  tan  6  .     rt. 

tan  2  6  =  - — (208) 

1  —  tan2  d  v       ' 

will  give  an  equation  from  which  4'  may  be  found. 
In  order  to  effect  this  elimination  more  easily,  make 

tan  4'  =  w, 
whence 

.     ,  u  ..  1 

sm  4^  =  -  ;  cos  4>  = 


yTT^2  yT+  u2 ' 

making  these  substitutions  above,  we  find 


2(F'u  4-  H')^/i  4-  w 
tan20  =  — 


A'  u2  4-  %lE'u  +  B'  —  G"(l  4-  m2) 


tan  4  = 


(Ff  -  H'  u)^/\  4-i 


is7'(l  —  U2)  4-  (.4'  -  i*')w 
which  in  Equation  (208)  give 

,         B'  F'-F'C'-E'H'    )  1 

{J?(i-«>)HA'-S')«}\.Hctir_A,N,+£rF>\  1  =0.  .  (209) 

-f  (jP'w  4-  H').(F'  -  H'uf     J 


MECHANICS    OF    SOLIDS.  157 

which  is  an  equation  of  the  third  degree,  and  must  have  at  least  one 
real  root,  and,  therefore,  give  one  real  value  for  ip.  This  value  being 
substituted  in  either  of  the  preceding  equations,  must  give  a  real 
value  for  0,  and  this  with  if),  in  either  of  the  Equations  (205)  or 
(206),  a  real  value  for  <f>;  whence  we  conclude,  that  it  is  always 
possible  to  assume  the  axes  so  as  to  satisfy  the  required  conditions, 
and  that  through  every  point  there  may  be  drawn  at  least  one  set  of 
principal  axes  at  right  angles  to  each  other. 

The  three  roots  of  this  cubic  equation  are  necessarily  real ;  and 
they  represent  the  tangents  of  the  angles  which  the  axis  x  makes 
with  the  lines  in  which  the  three  co-ordinate  planes  x'y',  y'z ',  x'z\  cut 
that  of  xy\  for  there  is  no  reason  why  we  should  consider  one  of 
these  angles  as  given  by  the  equation  rather  than  the  others,  and  the 
equations  of  condition  are  satisfied  when  we  interchange  the  axes 
x\  y',  z'.  Hence,  in  general,  there  exists  only  one  set  of  principal 
axes.  If  there  were  more,  the  degree  of  the  equation  would  be 
higher,  and  would,  from  what  we  have  just  said,  give  three  times  as 
many  real  roots  as  there  are  systems. 

If 

E'  =  H'  =  F'=zO, 

Equation  (209)  will  become  identical ;  the  problem  will  be  indetermi- 
nate, have  an  infinite  number  of  solutions,  and  the  body  consequently 
an  infinite  number  of  sets  of  principal  axes.  Such  is  obviously  the 
case  with  the  sphere,  spheroid,  &c.  . 

§  164. — Without  rotary  motion,  the  spherical  triangle  XNX  is 
invariable;  with  it,  the  point  N  has  a  motion  with  respect  to  X, 
X  to  iV,  and  Z'  to  Z.  These  points  being  on  the  surface  of  the 
unit  sphere,  their  relative  velocities  on  that  surface  arc  angular,  the 
first  about  the  axis  z,  the  second  about  the  axis  z\  and  the  third 
about  the  line  AN.     Taken  in  the  order  named  they  are, 

dxf)  d<f>t  dO 

It'  5F ;  di' 


158      ELEMENTS    OF    ANALYTICAL    MECHANICS. 

The  components  of  the  first  about  the  principal  axes,  x\  y\  z\  are, 
respectively,  Equation  (197), 


dtp 
~di 


.  sin  <f)  .  sin  6 ; 


of  the  second, 


dxb 

j-  .  cos  <f>  .  sin  6 ; 

at 

+  —  .cos0; 


0;  0; 


d(t>. 
di'1 


of  the  third, 


dd  dO      . 

-.cos^;  -^.sm^;         0. 


Taking  the  sum  of  those  about  the  same  axes,  there  will  result, 

dd  ,      dib      . 

v,  =  —  .  cos  0 —  .  sin  0  .  sin  0 ; 

dt  Y        dt  Y  ' 

dd      .    ^        di>  . 

vy  =  —  —  .  sin  0 -1-  .  cos  <p  .  sin  0 ; 

"■  =  di +  m  ■ cose: 

The  values  of  vx,  vf,  and  v„  being  found    from  Equations  (202)  and 

■■  ■ 
substituted  here,  and  the  resulting  equations  integrated,  will  give  three 

integral  equations  between  the  four  variables,  viz. :  0,  0,  0,  and  t.     The 

"latter    being    given,    the    former    become    known,    and    therefore    the 

position  of  the  principal  axes  will  be  determined. 

V   .The  principal  axes  through   the  centre  of   inertia,  having   the  same 

-    — •  i 

origin    as   those  drawn    in    a   fixed    direction,  will    describe  about  the 

latter,  as  axes,  surfaces  of  the  nature  of  cones,  more  or  less  irregular, 
depending  upon  the  shape  of  the  body  and  the  nature  of  the  im- 
pressed forces. 


MECHANICS    OF    SOLIDS.  159 


MOMENT     OF     INERTIA,     CENTRE     AND     RADIUS     OF     GYRATION. 

§  165. — The  quantities  A,  B,  and  C,  in  Equations  (201)  are  the 
moments  of  inertia  of  the  body  in  reference  to  the  principal  axes. 
To  find  these  moments  in  reference  to  any  other  axes  having  the 
same  origin  as  the  principal  axes,  denote  by  ■ 

x\  y\  z\  the  co-ordinates  of  m  referred  to  the  principal  axes;    by 

.r,    ?/,    z,    the    co-ordinates   of   the    same    element   referred   to    any 
other  rectangular  system  having  the  same  origin;   and  by 

C",  the  moment  of  inertia  referred  to  the  axis  z; 

then  from  the  definition, 

C  =  2m  (x2  -f-  y2)  =  2.mx2  -f-  I,my2) 

but  by  the  usual  formulas  for  transformation, 

x  =.  ax'   -j-  by'    -\-  cz\ 

y~ax    -f  by    -f  czy 

z  =  a  x  -f-  o  y  -|-  c  z , 

in  which  a,  b,  &c,  denote  the  cosines  of  the  angles  which  the  axes 
of  the  same  name  as  the  co-ordinates  into  which  they  arc  respectively 
multiplied  make  with  the  axis  corresponding  to  the  variable  in  the 
first  member. 

Substituting  the  values  of  x  and  y  in  that  of  C,  and  reducing 
by  the  relations, 

Imx'y'  =  0 ;         Smx'z'  =  0 *,         Smy  V  ac  0 ; 

and  we  have, 

C  =  a'2  .  1m  (y'2  +  z'2)  +  b"2  .  2m  (x2  -f  z2)  +  e"2 .  2m  (x'2  +  y'2)  ; 


160       ELEMENTS    OF    ANALYTICAL    MECHANICS, 
and  by  substituting  A,  B,  and   C  for  their  values,  this  reduces  to 

Cr  =  a"*A  +  b"*B  +  c"*C (210) 

That  is  to  say,  the  moment  of  inertia  with  reference  to  any  axis 
passing  through  the  common  point  of  intersection  of  the  principal 
axes,,  is  equal  .to  the  sum  of  the  products  obtained  by  multiplying 
the  moment  of  inertia  with  reference  to  each  of  the  principal  axes, 
by  the  square  of  the  cosine  of  the  angle  which  the  axis  in  question 
makes  with  these  axes. 
■ 

§  166. — Let  A  be  the  greatest  and  C  the  least  of  the  moments 
of  inertia,  with  reference  to  the  principal  axes;  then,  substituting  for 
a"2  its  value,  1  —  b"2 —  c"2,  in  Equation  (210),  we  have 

C'  =  A  —  b"2(A—B)—c"2(A—C)..     .     .     (211) 

By  hypothesis,  A  —  B  and  A  —  C  are  positive ;  therefore,  C  is 
always  less  than  A,  whatever  be  the  value  of  b"  and  c". 

Again,  substituting  for  c"2  its  value,  1  —  a"2  —  b"2,  in  Equation 
(210),  we  get 

C  =  C  +  a"2  (A  -  C)  +  b"2  (B-C)      ...     (212) 

and   C  must  always  be  greater  than   C. 

Whence,  we  conclude  that  the  principal  axes  give  the  greatest 
and  least  moments  of  inertia  in  reference  to  axes  through  the  same 
point. 

If  A  be  equal  to  By  then  will  Equation  (211)  become 

C'  =  (\—c"2)A  +  c"2C, (213) 

and  this  only  depending  upon  c",  we  conclude  that  the  moment  of 
inertia  will  be  the  same  for  all  axes  making  equal  angles  with  the 
principal  axis,  z\  The  moments  of  inertia,  with  reference  to  all  axes 
in  the  plane  x'y'  are,  therefore,  equal  to  one  another.  But  all  the 
axes   in    the    plane    x'y'    which    are   at   right   angles   to    one    another 


MECHANICS    OF    SOLIDS. 


161 


are,  §  164,  when  taken  with  z\  principal  axes,  and  we,  therefore, 
conclude  that  the  body  has  an  indefinite  number  of  sets  of  principal 
axes. 

If,  at  the  same  time,  we  have 

A  =  B  =  C, 

then  will  Equation  (210)  reduce  to 

a  =  C  =  A  =  B. 

That  is,  the  moments  of  inertia  are  all  equal  to  one  another,  and  all 
axes  are  principal,  the  Equation  (210)  being  satisfied  independently 
of  a",  6",  c". 


g  166*. — To  show  how 
the  moment  of  inertia  of 
any  body  varies  for  differ- 
ent positions  of  its  axis 
of  rotation,  07,  let  a,  (3,  y 
be  the  angles  made  by 
that  axis  with  the  co-ordi- 
nate axes.  Then  for  any 
one  of  the  particles,  m, 
whose  co-ordinates  are  *ar, 
y,  g,  we  shall  have 


or. 


r2  =  mn2  =  Om2  —  On2, 
r2  =  (x2  -+-  y2  +  z2)  —  (x  cos  a  +  y  cos  (3  -f-  z  cos  y)2. 


Squaring  and  reducing  by  the  relation 


1  =  cos2  a  +  cos2/3  -f  cos2  y, 


we  get, 


r2  =  (y2  +  z2)  cos2  «  +  (x2  +z2)  cos2/3  +  (x2  +  y2)  cos2y 

—  2xy  cos  a  cos  (3  —  2xz  cos  a  cos  y  —  2yz  cos /3  cos  y. 


3D62      ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Multiplying  by  m,  and  doing  the  same  for  all  the  elements  and  add- 
ing,  we  get, 

Hmr8  =  A  cos2«  -{-  B  cos2  (3  -f-  Ccos2y 

—  2i>cos«cosj3  —  2^cosacosy  —  2^008  0008  7; 

in  which 

D  =  I>mxyy  E  —  lmxzy  F  =  2  myz. 

Upon  the  axis  OyI,  supposed  to  assume  all  directions  through  0, 
take  the  length  0A>  or  point  A,  whose  co-ordinates  are  x',  y',  z'}  such 
that 

OA=—L=; 

\2nir2 

then 


cos  a  =  Jf'vSflM^ 


cos  ]3  =  y'^y/Smr2* 


cosy  =  z'V^r2? 

and  these  values  give    by  substitution,  after  suppressing   the    common 
factor, 

1  ss  ^#'2  +  %'2  +  6Y2  —  2Z>*y  —  2Ex'z  —  2Fy'z'. 

This  equation  gives,  for  the  locus  of  all  such  points  as  A,  a  sur- 
face of  the  second  order  with  its  centre  at  0.  The  radius  vector  OA 
eannot  be  infinite,  for  Smr2  cannot  be  zero ;  the  surface  is  therefore 
an  ellipsoid. 

This  ellipsoid  gives  a  clear  mental  conception,  or  perfect  geometric 
image,  for  the  ratios  of  all  the  moments  of  inertia  of  any  body,  with 
reference  to  different  axes  of  rotation  through  the  same  point.  For 
every  point  there  is  such  an  ellipsoid,  and  the  moments  of  inertia 
vary  as  the  reciprocals  of  the  squares  of  its  semi-diameters.  The 
ellipsoid  whose  axes  pass  through  the  centre  of  inertia  of  the  body 
is  called  its  central  ellipsoid. 


MECHANICS    OF    SOLIDS.  163 

If   we   take    for   co-ordinate    axes   those    which   are   the   principal 
axes  of  the  bodv,  then 

2mxy  =z  0,  2mxz  =.  0,  2myz  =  0, 

the  ellipsoid  becomes  central,  and  its  equation  is 

1  =  Ax'*  +  By'2  +  Cz'2 ; 

which  gives,  as  in  Equation  (210),  for  the  moment   of   inertia  about 
any  axis, 

Hmr*  =s  A  cos2  a  +  B  cos2  0  -f  6'  cos2  y. 

When  A  is  equal  to  2?,  the  ellipsoid  becomes  one  of  revolution; 

and  if  -? ,0- 

A  =  B  =  C 

the  ellipsoid  is  a  sphere. 

§  167. — Resuming   Equations  (33),  and  substituting  the  values  of 
x,  yy  z,  in  the  general  expression, 

2m  (x2  -f-  y2) 

which  is  the  moment  of  inertia  with  reference  to  any  axis  z,  parallel 
to  the  axis  z\  through  the  centre  of  inertia,  we  have 

2m  {*  +  y2)  =  2m  [ft  +  x'f  +  (y,  +  y')2] 

=  2m  (z'2  -f  y'2)  +  ft«  +  y2) .  2m 
-f  2#  .  2mx'  +  2y,  .  2my' ; 

but  from  the  principle  of  the  centre  of  inertia, 

2mx'  ss  0, 
and 

2my'  =  0 ; 

whence,  denoting  by  c?  the  distance   between   the  axes  z  and  z',  ana'  ' 
by  M  the  whole  mass, 

2m.(^+  y2)  ==  2m  (x'2  +  y'2)  +  Md2    .     .     .     (214) 


164  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

That  is,  the  moment  of  inertia  of  any  body  in  reference  to  a  given 
axis,  is  equal  to  the  moment  of  inertia  with  reference  to  a  parallel 
axis  through  the  centre  of  inertia,  increased  by  the  product  of  th« 
whole  mass  into  the  square  of  the  distance  of  the  given  axis  from 
that   centre. 

And  we  conclude  that  the  least  of  all  the  moments  of  inertia  is 
that  taken  with  reference  to  a  principal  axis  through  the  centre  of 
inertia. 

§  168. — Denote  by  r  the  distance  of  the  elementary  mass  m  from 
the  axis  z,  then  will 

r2  =  x2  +  y2, 
and 

2  m  (x2  -f-  y2)  =  2  ra  r2. 

Now,  denoting  the  whole  mass   by  M,  and  assuming 

2  m  r2  =  Mk'\ 


we   have 

(215) 


/2  m ) 


The  distance  k  is  called  the  radius  of  gyration,  and  it  obviously 
measures  the  distance  from  the  axis  to  that  point  into  which  if  the 
whole  mass  were  concentrated  the  moment  of  inertia  would  not  be 
altered.  The  point  into  which  this  concentration  might  take  place 
and  satisfy  the  condition  above,  is  called  the  centre  of  gyration. 
When  the  axis  passes  through  the  centre  of  inertia,  the  radius  k 
and  the  point  of  concentration  are  called  principal  radius  and  prin- 
cipal centre  of  gyration. 

The  least  radius  of  gyration  is,  Equation  (215),  that  relating  to 
the  principal  axis  with  reference  to  which  the  moment  of  inertia  is 
the  least. 

If  k4  denote  a  principal  radius  of  gyration,  we  may  replace 
2  m  (xf2  +  y'a)  in  Equation  (214)  by  Mk2,  and  we  shall  have 

Imr2  =  Mk2  =  M(k2  +  </*)      ....     (216) 


MECHANICS     OK     SOLIDS. 


165 


If  the  linear  dimensions  of  the   body  be  very  small  as   compared 
with  d,  we   may  write  the   moment  of  inertia  equal   to  Md2. 

The   letter   k    with    the    subscript    accent,   will   denote   a   principal 
radius  of  gyration. 

The  determination  of  the  moments  of  inertia  and  radii  of  gyratioh 
of  geometrical  figures,  is  purely  an  operation  of  the  calculus.  Such 
bodies  are  supposed  to  be  continuous  throughout,  and  of  uniform  den- 
sity. Hence,  we  may  write  d  M  for  m,  and  the  sign  of  integration  for 
2,  and  the  formula  becomes 


2  mr2 


=  fd  M. 


(21V) 


Example  1. — A  physical  line  about  an   axis   through  its  centre  and 

perpendicular    to   its    length. 

Denote   the  whole   length   by  2a;   then 
2  a  :  d  r  :  :  M :  d  Mi 


whence, 


and 


dr 
dM  =  M-—1 

2a 


Mk 


n — a  f& 

}  —    I     M  •  —  •  d  r  = 
1         «/<•  2a 


Ma2 
~3~; 


niinai  ' 


k.  = 


a 


•y/3 


If  the   axis  be   at  a   distance  d   from   the   centre,  and   parallel  to 
that   above,  then,  Equation  (216), 


k  =  -/Ja*  +  (P. 

Example  2. — A    circular   plate    of   uniform   density   and    thi;knet9% 
about   an  axis    through   its   centre   and  perpendicular   to  its  plane. 


166  ELEMENTS    OF     ANALYTICAL    MECHANICS. 


Denote  the  radius  by  a;  the  angle  XA  Q 
by  6 ;  the  distance  of  d  M  from  the  centre 
by  r;    then, 


whence, 


vro2  :  r .d&  .  dr     :  M\dM\ 


dM —  M-  — i — - — » 

or  a1 


and 


Mk, 


2  =  /    /     M.-^-d6  =   /    2M.  —  -dr  = 
J  o  v  o  tC  Or  v  o  a? 

*,  = a  VT> 

and  for  an  axis   parallel   to   the  above  at  the  distance  d, 


Ma* 


k  =  -v/£a2  +  d2. 

Example  3. — The  same  body  about  an  axis  through   its   centre  and 
in   its  plane. 


As  before, 


dM  =  M.r-dr:di, 


<x  a' 


in  which  r  denotes  the  distance  of  d  M  from  the  centre  ;  anchtaktng 
the  axis  to  be  that  from  which  &  is  estimated,  the  distance  of  che 
elementary  mass  from   the   axis  will    be   r  sin  0,  and 


J\f        pa    /»2T 


f*a    /»2  *         ... 3     s, , > 2  j)  M         pa    /»2  w 

Mk2  =        /     M  r-^—.dr.d6  =  -^—        /     r*(]  -cos2  6)  dr.dt, 


Mk. 


M    Pa  a 

2  =  ^—-  I    r3 .  dr  =  M  —i 

1  a2  J  o  4 


and 


kt  =  i  a, 
and  about  an   axis  parallel  to  the  above  and  at  the  distance  rf. 


vl" 


2  +  rf2. 


MECHANICS    OF     SOLIDS. 


167 


It  is  obvious  that  both  the  axes  first  considered  in  Examples  2 
and  3  are  principal  axes,  as  are  also  all  others  in  the  plane  of 
the  plate  and  through  the  centre,  and  if  it  were  required  to  find 
the  moment  of  inertia  of  th«  plate  about  an  axis  through  the  centre 
and  inclined  to  its  surface  under  an  angle  <p.  the  answer  would  be 
given  by  the  Equation  (210), 

Mk2  —  £  Ma2  sin2  <p  -h  y  Ma2  cos2  <p 
—  \Ma2{\  -f  sin2cp), 

and  for  a  parallel  axis  whose    distance  is  d, 

Mk2  =  M  (  i   a2  (1  -I-  sin2  9)  +  d?)  • 

Example  4. — A  .solid  of  revolution  about  any  axis  perpendicular  to 
the   axis   of  the   solid. 

Let  D  A*  E  be  the  given  axis, 
cutting  that  of  the  solid  in  A'>  Let 
A'  be  the  origin  of  co-ordinates, 
PM=y\  A'P=zx;  AA'  =  m; 
A'  B  =  n  ;  and  V  =  volume  of  the 
solid. 

The  volume  of  the  elementary 
section  at  P  will  be 


«r  y2  .dx, 


and 
whence, 


V  :  M :  :  if  .y2  .dx  :  d M\ 


d  M  =  —  •  if  •  y2  •  d  x, 


and  its  moment  of  inertia  about  MM',  is,  Example  8, 

M  y2 

—  •  *  -  v*  •  a  x  •  --  • 
V         y  4 

and  about  the  parallel  axis,  D  E, 

M 


y.«.y*.dx(\y2  +  x2) 


168  ELEMENTS    OF     ANALYT.CAL    MECHANICS, 

therefore, 

J  m     v 


But 


whence, 


v  =f**y***i 


fm   (i2/4  +  *2r).dx 

k2  =  — 


m 


X 


The  equation   of  the   generating  curve   being   given,  y  may  be  elimi- 
nated  and  the  integration  performed. 

Example   5. — A  sphere   about   a   line    tangent   to  its   surface. 
The  equation  of  the  generatrix  is 

y2  =  2  a  x  —  x2 ; 

in  which  a  is  the  radius  of  the  sphere.     Substituting   the  value  of  y* 
in  the  last  equation,  recollecting  that   m  =  0,   and  n  =  2  a,  we  have 

f  °(a2  x2  +  ax3  —  |  x4)  dx 

k2  —  — ==  -s  a2- 

/•2«  5 

J      (2  ax  -  x2)dx 

Also  Equation  (216), 

IM  =  k2  -  a2  =  %  a2, 

'  5 

and 


Thus,  when  the  boundary  of  a  rotating  body  an  J  the  law  of  its 
density  may  be  defined  by  equations,  its  moment  of  inertia  is  readily 
found  by  the  ordinary  operations  of  the  calculus ;  but  when  the  figure 
is  irregular  and  the  density  discontinuous,  recourse  is  had  to  the  prop- 
erties of  the  compound  pendulum,  to  be  explained  presently. 


MECHANICS     OF    SOLIDS.  169 

Example  6. — Find  the  points  in  reference  to  which  the  principal  mo- 
ments are  equal. 

Take  the  origin  at  the  centre  of  inertia,  and  the  principal  axe? 
through  that  point  as  the  co-ordinate  axes.  Denote  by  xt  yt  z4  the  co 
ordinates  of  one  of  the  points  sought ;  by  A4,  Bt,  and  Ct  the  principal 
moments  with  reference  to  this  point,  and  by  x  y'  z'  the  co-ordinates  oi 
the  element  m.  Then,  because  the  moments  through  the  point  xt  yi  zt 
are  to  be  principal,  will 

2m(x'-x4)(y'-y()  =  Q;  2m(x'-xt)  («'-z,)  =  0;  2m(/-^J(z'-2/)  =  0. 

Performing  tlie  multiplication  and  reducing  by  the  properties  of  the 
centre  of  inertia  and  principal  axes,  we  have 

M .  xt  yt  =  0  ;   Mxt  zt  —  0  ;    Myt  zt  =  0  : 

which  can  only  be  satisfied  by  making  two  of  the  co-ordinates  xf  yt  zt 
separately  zero.     Let  yt  —  0;  and  zt  —  0  ;    then,  §  167  and  Eq.  (216). 

A4  =  A;    Bt  =  B  +  Mxy,    C4=C+Mx4*; 

but,  by  the  conditions,  the  first  members  are  equal.     Whence 

A  =  B  +  M  x;  =  C  +  Mx4*  ; 
and,  therefore, 

B=C;    and  ^  =  ±>/-^?; 

and  from  which  it  is  apparent :  1st,  that  if  all  the  principal  momenta 
in  reference  to  the  centre  of  inertia  be  unequal,  there  is  no  point  in 
reference  to  which  they  can  be  equal ;  2d,  that  if  two  of  them  be 
equal  in  reference  to  the  centre  of  inertia  and  the  third  be  the  great- 
est, there  are  two  points,  equally  distant  from  the  centre  of  inertia  and 
on  the  axis  of  the  greatest  moment,  with  reference  to  which  they  are 
equal;  3d,  that  if  all  three,  with  reference  to  the  centre  of  inertia,  be 
equal  to  one  another,  there  is  no  other  point  with  respect  to  which 
they  can  be  equal. 

IMPULSION. 

§  169. — We  have  thus  far  only  been  concerned  with  forces  whose 
action  may  be  likened  to,  and  indeed  represented  by,  the  pressure 
arising   from   the  weight   of   some    defiuite   body,  as   a   cubic   foot   of 


170         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

distilled  water  at  a  standard  temperature.  Such  forces  are  called 
incessant,  because  thev  extend  their  action  through  a  definite  and 
measurable  portion  of  time.  Such  a  force  is  assumed  to  be  measured 
by  the  whole  effect  which  its  incessant  repetition  for  a  unit  of  time 
can  produce  upon  a  free  body.  The  effect  here  referred  to  is  called 
the  quantity  of  motion,  being  the  product  of  the  mass  into  the 
velocity  generated.     That  is,  Equations  (12)  and  (13), 

P  =  M.rt  =  Md^.=Jf^;    ....   (218) 

ill  which    Vt,  denotes  the  velocity  generated  in  a  unit   of  time. 

The  force  P,  acting  for  one,  two,  or   more    units   of  time,    or    for 
any  fractional  portion  of  a  unit  of  time,  may  communicate  any  other 
velocity    F,  and  a  quantity  of  motion    measured   by    M  V.      And    if 
the  body  which  has  thus  received  its  motion  gradually,  impinge  upon 
another    which    is    free    to    move,    experience    tells    us    that    it    may 
suddenly    transfer    the    whole   of    its    motion    to    the   latter   by    what 
seems  to  be  a  single  blow,  and  although   we    know   that   this  transfer 
can  only  take  place   by  a   series   of  successive   actions   and   reactions 
between    the    molecular   springs   of  the    bodies,  so    to   speak,  and  the 
inertia  of  their  different  elements,  yet  the  whole  effect  is  produced  in 
a  time  so  short  as  to  elude  the  senses,  and  we  are,  therefore,  apt   to 
assume,    though   erroneously,    that   the    effect   is   instantaneous.      Such 
an  assumption  implies  that  a  definite  velocity  can  be  generated  in  an 
indefinitely  short  time,   and  that  the  measure  of   the   force's   intensity 
is,  Equation  (218),  infinite. 

In  all  such  cases,  to  avoid  this  difficulty,  it  is  agreed  to  take  the 
actual  motion  generated  by  these  blows  during  the  entire  period 
of  their  action,  as  the  measure  of  their  intensity.  Thus,  denoting 
the  mass  impinged  upon  by  J/",  and  the  actual  velocity  generated 
in    it   when   perfectly    free   by    T,   we   have 

P  =  MV    =  M.p(, (219) 

in   which   P,    denotes  the    intensity    of    the     force's    action,    and    the 
second  member  of  the  equation  the  resistances  of  the  body's  inertia 


MECHANICS    OF    SOLIDS 


171 


Forces    which   act   in   the    manner   just   described,  by    a   blow,  are 
sometimes  called  impulsive,  and  their  action  impulsion. 

MOTION     OF     A     BODY     UNDER     IMPULSION. 

§  170. — Th*  components  of  the  inertia  in  the  direction  of  the  axes 
xyz.  are  respectively 

IL    ds   dx        w    dx 

M-- —  as  M  --—  ; 

d  t   as  d  t 

__   ds   dy  dy 

d  t   ds  d  t 

_,  ds   dz        j*    dz 

M'  —  '—  =  M-~; 

dtds  d  t 

which,    substituted   for    the    corresponding    components   of   inertia   in 
Equations  (B)  and  ((7),  give 

dx    1 


2  P  cos  a  =  2  to  • 


dt 


2  Pcos/3  =  2to-  -?; 

a  £ 

7 

2  P  cos  r  =  2  to  •  —  ; 

1  dt     J 


•     •     • 


(220) 


2  P(r'  cos  j8  —  y'  cos  a)  =  2  to  (xf  ~  —  y'  •  -^J  , 

* ,  .  ,  n  /  /     dx         .    dz\ 

2  P  {zf  cos  a  —  #'  cos  7)  =  2  to  (  z '  •  —  —  x  •  —  )  , 

2  P  (/ cos y  —z'  cos  /3)  =  2  to  (y'  •  -77  —  2'  •  ~)  • 


y  .      (221) 


In  which  it  will  be  recollected  that  x  y  z  are  the  co-ordinates  of  to, 
referred  to  the  fixed  origin,  and  x'  y'  zr,  those  of  the  same  mass 
referred  to  the  centre  of  inertia. 


MOTION    OF   THE   CENTRE     OF    INERTIA. 

§  1*71. — Substituting    in    Equations    (220),    for    dx,    dy,  d  z,    their 

values   obtained    from  Equations  (34),  and  reducing  by  the  relations 

2mdx'  =  Q;    2*idy'  —  Q\  2mdz'  =  0\   •  •        (222) 


172         ELEMENTS    OF    ANALYTICAL    MECHANIC8 
given  by  the  principle  of  the  centre  of  inertia,  we  find 


dx. 


2  P  cos  a  =  -—  .2m; 

2Pcos/3 

2  P  cos  y 


dt 

dt  ' 


> 


dz 


dt 


and  substituting  M  for  2  m,  we   have 


-•2m; 


_,  ,  ,   d  x . 

2PcOSa    ZS.   M'—r^', 

dt 


2Pcos/3  =  jlf  .^-'; 

^  dt 


-.*    d  z. 
2  P  cos  y  =  M  •  -t-^  ; 

1  dt  ' 


•        k 


(223) 


which  are  wholly  independent  of  the  relative  positions  of  the  elements 
of  the  body,  and  from  which  we  conclude  that  thr.  motion  of  the 
centre  of  inertia  will  be  the  same  as  though  the  mass  were  concen- 
trated in    it,   and  the   forces   applied  immediately  to  that  point. 

§  172. — Replacing  the  first  members  of  the  above  equations  by 
their  values  given  in  Equations  (41),  and  denoting  by  V  the  velocity 
which  the  resultant  R    can  impress  upon  the  whole  mass,  then  will 

2  P  cos  a  =  M  V  cos  a ;    IP  cos  ft  =  M  V  cos  b  ;    2  P  cos  y  =  M  V  cos  c  ; 


substituting   these   above,  we  find 

V.  cos  a 
V . cos  b 
F.oos  c 


~    dt   ' 

-  iii . 

dt    ? 

dzt 
=  17  ' 


(224) 


MECHANICS    OF    SOLIDS. 


173 


Mid  integrating, 

xt  =  V-  cos  a .  t  -f  C,     ' 

y,  =  F.cos&.*  -f  C",    I     .     .     .     .     .     (225) 

zt  =  V.  cos  c.t  +  C",  4 

and   eliminating   f   from   these   equations,    V  will   also  disappear,    and 
we  find, 

cos  c 


z, 

1 

xr 

z< 

= 

y,- 

y* 

xr 

cos  a 

cos  c 
cos  6 

cos  b 
cos  a 


C" 

cos 

c   — 

<?'" 

cos  a 

cos  a 

; 

C" 

cos 

c  — 

cm 

cos 

b 

cos 

b 

■  J 

c 

cos 

b  - 

G" 

cos 

a 

— ? 

cos  a 


(226) 


which  being  of  the  first  degree  and  either  one  but  the  consequence 
of  the  other  two,  are  the  equations  of  a  straight  line.  This  line 
makes  with  the  axes  a?,  y,  z,  the  angles  a,  b,  c,  respectively,  and  b>T 
therefore,  parallel    to    the   resultant  of  the  impressed  forces. 

Whence  we  conclude,  that  the  centre  of  inertia  of  a  body  acted 
upon  simultaneously  by  any  number  of  impulsive  forces,  will  move 
uniformly  in  a  straight  line  parallel  to   their   common   resultant. 


MOTION    ABOUT   THE   CENTRE   OF   INERTIA. 

§  173. — Substituting,  in  Equations  (221),  for  dx,  dy  and  dz,  theii 
values    from    Equations    (34),    reducing   by 

2mx'  =  0, 
2  m  y'  =  0, 

and  we  find, 

2  P  (x'  cos  (3  —  y'  cos  a)  =  2  m  (x'  •  -~   —  yr  —^-J  ; 

2  P(z'  cos  a  —  x'  cos  y)  —  2  m  (zf  •  -^- *'  •  (-^-J  ;    [    •  .  (227) 

2  P  (/  cos  7  —  z'  cos  B)  —  2  m  (y'  •  -A    -  z'  •  -j-  J  ; 


174  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

whence,  the  motion  of  the  body  about  its  centre  of  inertia  will  be 
the  same  whether  that  point  be  at  rest  or  in  motion,  its  co-ordinates 
having   disappeared   entirely  from  the  equations. 

ANGULAB   VELOCITY. 


§  174. — Replacing  the  first  members  of  Eqs.  (227)  by  £ ,  Mr  and  Nn 
respectively,  §  162  ;  and  substituting  in  the  second  members  fur  dx\  dy' 
and  dz\  their  values  in  Eqs.  (190),  we  readily  find 


d$ 
dt 

~dt 


dm 
~d~t 


da  d-L 

Zy-f-2  m  x'z  •  — -f  2  m  v'z'  •  — 


2  m  {x"z  +  y"1) 


dt 


M^mz'y'.-^+Zmx'y'.^ 


2  m  (x'2  +  z'2) 
'  *       dt  dt 


(228) 


2  m  (f*  +  z'2) 

If  the  axes  be  principal,  then  will  2inx'y'  =  0,  2  m  y V  =  0, 
2  m  zfz'=0',  or  if  the  axes  be  fixed  in  succession,  then  for  the  axis  x'  will 
dip  =  0;  dp  =  0;  for  the  axis  y,  dcp  =  0;  c/g)  =  0;  and  for  the  axis 
z,  d-a  =  0;  dip  =  0,  and  the  above  become 


c?0  Ijt 

Vz  ~  It   ~  £t»  ..(*?.+.**)  :     C 


iir. 


Vy  ~  "^7  —  2  m .  (jftfl 2)  :      B  ' 


v„ 


c?cr 
tf7 


iV 


y 


i . 


2«*.(y8  +  z2)        A  \ 


(220) 


That  is,  the  component  angular  velocity  about  either  a  principal  or  fixed 
axis,  is  equal  to  the  moment  of  the  impressed  forces  di\ided  by  the 
.-moment  of  inertia  with  reference  to  that  axis. 


The    resultant    angular    velocity    being    denoted    by 
have,  (Eq.   196), 


dst 


^  =  i-  WV  +  rf+2  +  dvfl. 
at         a  t  V 


we    also 


(230) 


MECHANICS    OF    SOLIDS.  175 

AXIS    OF    INSTANTANEOUS    ROTATION. 

§  175. — The  axis  of  instantaneous  rotation  is  found  as  in  §.153,  by 
making,  in  Equations  (192),  d  x  =  0,  c?y'=  0,  dz'  —  0;  and,  therefore, 

z' .  vy  —  y' .  vt  =  0  ;    x' .  vz  —  z' .  vx  =  ()  ;    y' .  vx  —  x  .  vy  =  0   .  (23 1 ) 

which,  as  the  last  is  but  a  consequence  of  the  others,  are  the  equations 
of  a  right  line  through  the  centre  of  inertia. 

The  equations  of  the  line  of  the  resultant  impact  are,  Eqs.  (45), 

—  x  • *  ***  X  '  y  ~  X  '  ■    ~  x  ; 

and  the  inclination  d  of  this  line  to  the  instantaneous  axis,  is  given  by 

.              vt.Z  +  vu.Y+vx.X 
cos  a  =  — 


Vv:  +  v;  +  v: .  vz*  +  y  +  x* ' 

or,  substituting  for  i>2,  vy,  and  vx  their  values,  Eqs.  (229)  and  (191), 

l  .  z    if ; .  r    ir .  x 

-L 1 i j £__- 

cos  0  = __.° B       _ .     (232) 

AW*  ©>  (§) '■  <*?*£»» 

The  point  in  which  the  line  of  the  impact  pierces  the  plane  yz  is  given  by 

9  —  x  »  y  ~       X  ' 

dividing  one  by  the  other,  we  have,  for  the  equation  of  the  line  through 
this  point  and  the  centre  of  inertia, 

Denote  the  angle  which  this  line  makes  with  the  instantaneous  axis  by 
6' ;    then  from  the  equations  of  these  lines  will 

cos  a  =  — — • ; 

2 


AWW-A-W 


+  1 


or,  Eqs.  (229)  and  (191),  ^ 


cos0'  = r=    •     I238) 


176      ELEMENTS    OF    ANALYTICAL    MECHANICS. 


AXIS     OF     SPONTANEOUS     ROTATION. 

§  176. — If  both  members  of  Eqs.   (34)  be  divided  by  dt,  we  have 

dx       dxt        dxr  dy       dyt       dt/  dz       dzt       dz' 

dt'     ~di  + '  ~~di " '  dt  '     ~di  +  ~dt  '         It" dt "  ~*~ di " ' 

,  .„  -  ,  dx  dy  dz  .  . 

and  if  for  any  element  —  =0;     —  =z  0 ;     —  =  0  .     .     .     .     (234) 

...    dxt  dx'       dyt  dy'      dz  dz 

then  will  -tt  =  —  -77  ;     ~§i  = TV ;     -r1  =  —  ~r  •     •     •     (235) 

eft  dt        dt  dt'     dt  dt  v       ' 

Substituting  for  the  first  members  their  values  given  in  Equations  (224), 

and  for  the  second  members  their  values  given  in  Equations  (192),  we 

have  z'  .  vy  —  y' .  vz  +  V  .  cos  a  =  0 

x' .  v,  —z'  .  vx  +  V.cosb  =  0   I    .     .     .     .     (236) 

y' .  vx  —  x' .  vy  -f-  V .  cos  c  =0 

Now,  if  either  of  these  equations  be  but  a  consequence  of  the 
other  two,  then  will  they  be  the  equations  of  a  right  line  parallel, 
Equations  (231),  to  the  instantaneous  axis;  and  all  points  upon  this 
fine  will  be  at  rest  during  the  body's  motion. 

This  line  is  called  the  axis  of  spontaneous  rotation  ;  because,  being 
at  rest,  Equations  (234),  while  the  centre  of  inertia  is  in  motion,  the 
whole  body  may  be  regarded,  during  impact,  as  rotating  about  this 
line.  Its  position  results  from  the  conditions  of  Equations  (235), 
which  are,  that  the  velocity  of  each  of  its  points  and  that  of  the 
centre  of  inertia  must  be  equal  and  in  contrary  directions.  The  dis- 
tinction between  the  axes  of  instantaneous  and  of  spontaneous  rota- 
tion is,  that  the  former  is  in  motion  with  the  centre  of  inertia,  while 
the  latter  is  at  rest. 

To  find  the  conditions  which  shall  express  the  dependence  of  either 
of  the  Equations  (236)  upon  the  other  two,  multiply  each  by  the 
angular  velocity  it  does  not  already  contain,  add  the  products,  and 
divide  the  sum  by  the  resultant  angular  velocity  vt\   there  will  result, 

V  V  V 

cos  a  .  —  +  cos  &.  —  -{-  cos  c  .  — -  =  0  .     .     .     (236)' 
v-  v-  v 

vx  wi  "t 

The  first  member  is  the  cosine  of  the  an<xle  which  the  resultant 
impact  makes  with  the  instantaneous  axis:  this  being  zero,  it  follows 
that  whenever  a  body  is  struck  so  as  to  make  the  instantaneous  axis  per- 
pendicular to  the  direction  of  the  impact,  the  spontaneous  axis  will  exist. 


MECHANICS    OF    SOLIDS.     '  177 

Denote  by  Z,  the  distance  from  the  spontaneous  axis  to  the  line  of 
the  impact;  by  et  and  <?y,  the  absolute  terms  in  the  second  and  third. 
of  Eqs.  (236),  solved  with  respect  to  z  and  y,  then  will 


,  (*-?)C!-a+(*+3)(H) 

V(yv@w(!)vrov. 


(237) 


Equation  (232)  will  make  known  the  circumstances-  of  the  impact 
and  shape  of  the  body  which  will  determine  the  existence  of  the  spon- 
taneous axis. 

Make  the  impact  in  the  plane  of  the  principal  axes  x'  y\  and  pav 
allel  to  the  axis  x\  Then,  Equation  (232),  will  6  =  90°,  and  the  spon- 
taneous axis  will  exist  Also,  Equation  (233),  6'  =  90°.  And,  Equation* 
(237)  and  (229),  and  because  X=  M.  V,  and  V=ey.vz, 


i  =  ey-t- 
whenee, 


*  —  *y  -f-   xr  —  ey  -J-    , f  —  <V  "      1   » 


M .  ey  .  vz  y        ey 


(l-ev).ey  =  lc? (23d). 

That  is,  when  the  line  of  the    impact  is  in   the   plane  of   two  of  the 
principal  axes  and  parallel  to  one  of  thein,  there  will  be  a  spontaneous 
axis,  and  the  product  of  its  distance  from  the  centre  of  inertia  by  that, 
of  the  line  of  the  impact  from  the  same  point,  is  equal  to  the  square 
of  the  principal  radius  of  gyration  in  reference  to  the  instantaneous  axis. 

§  17*7. — The  body  being  free,  and  the  axis  of  spontaneous  rotation  at 
rest,  while  the  other  parts  of  the  body  are  acquiring  motion,  the  forces, 
both  extraneous  and  of  inertia,  are  so  balanced  about  that  line  as  to  im- 
press no  action  upon  it.  The  line  of  the  impact  and  the  points  of  the 
body  on  this  line  are  called,  respectively,  the  axis  and  centres  of  percussion, 
in  reference  to  the  spontaneous  axis.  A  centre  of  percussion  in  refer- 
ence to  an  axis  is,  therefore,  any  point  at  which  a  body  may  be  struck 
without  communicating  a  shock  to  a  physical  line  coincident  in  position 
with  that  axis. 

STABLE    AND    UNSTABLE    ROTATION. 

§  178. — Now  suppose  the  rotation   to   have  been   impressed,  the  in- 


1*78     ELEMENTS  OF  ANALYTICAL  MECHANICS. 

stantaneous  axis  nearly  coincident  with  the  principal  axis  z,  and  the  body 
abandoned  to  itself.     What  will  be  the  circumstances  of  the  motion  ? 

The  first  member  of  the  third  of  Equations  (194)  will  be  sensibly 
equal  to  unity,  vy  and  vx1  therefore,  indefinitely  small,  their  product  an 
indefinitely  small  quantity  of  the  second  order;  Z  ,  Jf  ,  and  Nt  will  be 
zero,  and  Equations  (202)  may  be  written, 

6'.^'  =  0;  B.^  +  vx.vx.(A-C)  =  0;  A.-g  +  v, .  v,.(C-B)=o. 

Integrating  the  first,  we  have 

v,  =  77  =  »; 
in  which  c  is  the  constant  of  integration ;  and  this  in  the  other  equations 
gives 

B.d£  +  n.(A-G).vx  =  0;    A.^f  +  n(C-B)vy  =  0; 

differentiating   and   substituting    in   each   of  the  derived   equations  the 

values  of  the  first  differential  coefficients   obtained    from  the  primitive, 

we  find 

d>v  (A-C).(B-C) 


d  t*  A.B 

<?vm        .   (A-C).(B-C) 


.vx  =  0. 


dT+U'  A.B  ~        ' 

If  A  —  G  and  B  —  C  be  both   positive  or  both  negative,  their  product 
will  be  positive,  and  the  integrals  are 


v,  =  ay .  sin  |  n  .  \  * -j-^ -7  .  t  +  cf  J ; 


va  —  am .  sin  | n  .  y  - '— ^ .  t  +  c,\. 

If  one  of  the  factors  A  —  G  and  B  —  G  be  positive  and  the  other  neg- 
ative, their  product  will  be  negative,  and  the  integrals  will  be 


vf  =  ay .  e         v  *'? 

vx  —  ax.e  V  a.b 

In  these  integrals,  oy,  am,  cy,  and  cx,  are  constants  whose  values  result 
from    the    initial    conditions   of   the    rotation.     They  are   small    at    the 


MECHANICS    OF    SOLIDS 


170 


epoch,  because  v9  and  vt  are  small.  In  the  first  integration,  vy  and  vt 
will  continue  small  and  resume  periodically  their  initial  values;  in  the 
second,  they  will  increase  with  the  time  indefinitely.  If  the  instantane- 
ous axis  coincide  with  the  axis  z,  then  will  ay  and  am  be  zero;  v9  and 
rx  will  be  zero,  and,  hence,  a  principal  axis  is  always  a  permanent  hut* 
of'  rotation ;  and  the  rotation  will  be  stable  about  the  axes  of  greatest 
and  least  moments  of  inertia,  and  unstable  about  all  other*. 

MOTION   OF   A    SYSTEM   OF   BODIES. 


§179 — We  have  seen  that  the  Equations  (117)  and  (119)  give 
all  the  circumstances  of  motion  of  the  centre  of  inertia  of  a  single 
body  in  reference  to  any  assumed  point  taken  as  an  origin  of  co 
ordinates.  For  a  second,  third,  and  indeed  any  number  of  bodies, 
referred  to  the  same  origin,  we  would  have  similar  equations,  the 
only  difference  being  in  the  values  of  the  co-ordinates,  of  the  inten 
sities  and  directions  of  the  forces,  and  of  the  magnitudes  of  the  masses. 
This  difference  being  indicated  in  the  usual  way  by  accents,  we  should 
obtain  by  addition, 


2  M- 


2  M- 


2  M  • 


dt2 

d*z 
dt2 

d*z 


=  2X; 


=  2  F: 


=  2Z; 


(239) 


»jr(ir.girig)i.»<jr;-xf), 

)  =  2  (X  z  -  Z  x)  ; 


XM 


'2x 


/      d2 
2  3/  (y. 


d2z 


dP 


—  x 


—  z 


d2Z 

~dfi 


#K*W-*4l 


(240) 


in  which  it  must  be  recollected  that  x,  y,  z,  &c,  denote  the  co 
ordinates  of  the  centres  of  inertia  of  the  several  masses  Af,  &c* 
referred    to   a   fixed  origin. 


180 


ELEMENTS    OF     ANALYTICAL    MECHANICS. 


MOTION    OF    THE    CENTRE   OF   INERTIA    OF   THE    SYSTEM. 

§180. — Taking  a  movable  origin  at  the  centre  of  inertia  of  the 
wtire  system;  denoting  the  co  ordinates  of  this  point  referred  to 
Jie  fixed  origin  by  *,..  yt ,  *it  and  the  co-ordinates  of  the  centres 
of  inertia  of  the  several  masses  referred  to  the  movable  origin  by 
x\  y',  z\  &c,  we  have,  the  axes  of  the  same  name  in  the  two  sys- 
tems being  parallel, 


and 


x  =  xt  -\-  x\ 

y  =  y<  +  y', 

z  =  zt  +  z\ 

cPx 

=  d2x,  +  <P  x', 

&y 

=  d*  y,  +  d*  y\ 

d2z 

=  d*z,  +  d*  z',  \ 

(241) 


which  substituted   in  Equations  (239),  and  reducing  by   the  relations 
2Jf.cZ2*'  =  0;     2  Md2 y'  =  0 ;     2Jfo?22/  =  0;    ■    .(242) 
obtained  from   the  property  of  the   centre  of  inertia,  we  find 


d2x 


dP 

dp 

cPz 


dp 


•  2M=  2  T 
i.xM=2  Z 


(243) 


which  being  wholly  independent  of  the  relative  positions  of  the  several 
bodies,  show  that  the  motion  of  the  centre  of  inertia  of  the  system 
will  be  the  same  as  though  its  entire  mass  were  concentrated  in 
that  point,  and   the  forces  applied  directly  to  it. 


§  181. — Multiplying  the  first  of  Equations,  (243),  by  yt,  the  second 


MECHANICS     OF     SOLIDS. 


181 


y/.g).2¥=V2r-V2Z; 


by  xt,  and  taking  the  difference ;  also,  their  first  by  zt  the  third 
by  x„  and  taking  the  difference,  and  again  the  second  by  z4t  the 
third   by  y4,   and   taking    the   difference,   we  find 

(z   th 

(d2  r  d2  z\ 

z'--dfi-x'-d^)-sMz=z'-:EX-x'-:EZ-  >•  (244> 

which  will  make  known  the  circumstances  of  motion  of  the  common 
centre    of  inertia   about   the  fixed   origin. 

MOTION   OF   THE   SYSTEM    ABOUT   ITS   COMMON   CENTRE   OF   INERTIA. 

§  182. — Substituting  the  values  of  x,  y,  z,  d2  x,  &c,  given  by 
Equations  (241),  in  Equations  (240)  and  reducing  by  Equations  (244) 
and    (242),    there    will    result 


*(/^-^-*V'-r') 


(245) 


Equations  from  which  all  traces  of  the  position  of  the  centre  of 
inertia  have  disappeared,  and  from  which  we  conclude  that  the 
motion  of  the  elements  of  the  system  about  that  point  will  be  the 
same,  whether  it  be  at  rest  or  in  motion.  These  equations  are 
identical  in  form  with  Equations  (118);  whence  we  conclude  that 
the  molecular  forces  disappear  from  the  latter,  and  cannot,  there 
fore,  have  any  influence  upon  the  motion  due  to  the  action  of  the 
extraneous  forces. 

CONSERVATION   OF   THE   MOTION   OF   THE  CENTRE    OF   INERTIA. 

§  183. — If  the  system  be  subjected  only  to  the  forces  arising  from 
the    mutual    attractions   or    repulsions  of  its    several    parts,    then   will 

2  .Y  =  0 ;  27=0;  2  Z  =  0. 


182         ELEMENTS    OF     ANALYTICAL    MECHANICS. 

Fcr,  the  action  of  the  mass  J/",  upon  a  single  element  of  M, 
will  vary  with  the  number  of  acting  elements  contained  in  M\ 
and  the  effort  necessary  to  prevent  M'  from  moving  under  this 
action  will  be  equal  to  the  whole  action  of  M  upon  a  single  element 
of  M'  repeated  as  many  times  as  there  are  elements  in  M'  acted 
upon  ;  whence,  the  action  of  M  upon  M'  will  vary  as  the  product 
MM.  In  the  same  way  it  will  appear  that  the  force  required  to 
prevent  M  from  moving  under  the  action  of  M\  will  be  propor- 
tional to  the  same  product,  and  as  these  reciprocal  actions  are 
exerted  at  the  same  distance,  they  must  be  equal ;  and,  acting  in 
contrary  directions,  the  cosines  of  the  angles  their  directions  make 
with  the  co-ordinate  axes,  will  be  equal,  with  contrary  signs.  Whence, 
for  every  set  of  components  P  cos  a,  P  cos  /3,  P  cos  y,  in  the 
values  of  2  J,  2  F,  2  Z,  there  will  be  the  numerically  equal  com- 
ponents,  —  P'  cos  a',  —  P1  cos  /3',  —  P'  cos  y\  and,  Equations  (243), 
reduce,   after   dividing   by   2  M,    to 

and   from  which  we   obtain,  after   two  integrations, 

x4=  C  .t  +  D';    ^ 

yt  =  C".t  +  £>";    I •     (247) 

zt  =  <7'".*  +  D'";  J 

in  which  C",  C'\  C"\  D\  D"  and  D'"  are  the  constants  of  inte- 
gration ;  and  from  which,  by  eliminating  /,  we  find  two  equations  of 
the  first  degree  between  the  variables  xt ,  y, ,  z; ,  whence  the  path 
of  the   centre   of  inertia,  if  it   have  any  at   all,  is   a   right   line. 

Also    multiplying    Equations  (246)   by  2dxf,  2dyt,  2dZj,  respec- 
tively, adding  and    integrating,  we   have 

««,»  +  W  +  i;*  g  y»  =  0   ....   (248) 

in  which  C  is  the  constant  of  integration  and  V  the  velocity  of  the 
centre  of  inertia  of  the  system.  From  all  of  which  we  conclude, 
that    when   a  system    of    bodies    is    subjected    only  to    forces   arising 


MECHANICS    OF    SOLIDS. 


183 


from  the  action  of  its  elements  upor.  each  other,  its  centre  of  inertia 
will  either  be  at  rest  or  move  uniformly  in  a  right  line.  This  is 
called   the   conservation  of  the  motion  of  the   centre  of  inertia. 


CONSERVATION     OF    AREAS. 

§184.— The    second   member   of  the  first  of  Equations  (215)  may 
be  written, 

rv  -  xy  +  r  f\  -  xyn  +  &c. ; 

and    considering   the   bodies   by  pairs,  we  have 

X  =  -  X' ;     Y  =  -  V  ; 
and   eliminating  X  and   Y'  above  by  these  values,  we   have 

y  (*'  _  x»)  _  X(y'  -  y")  4-  &c. 
But, 


JT=  P. 


a'  -  *" 


;      Y  =rP 


y'  ~  y" 


p     '  p 

in  which  p  denotes  the    distance    between    the   centres   of    inertia   o( 
the  two  bodies.     And  substituting  these  above,  we  get 

p  .   I L.  (*'    _   *")   -  P  • —  (y'    _   y")    -=    0  ; 

and   the   same   being   true   of  every    other  pair,  the  second    members 
of  Equations  (245),  will    be  zero,  and    we   have 


M    /  ,    d2x'         ,    d2z\ 


and  integrating 


M     /  ,    d2zr  ,    d2y'\ 

2  M  •  (  y' Z*  •  — -  )  —  0 

V      dt2  rf*2/ 


dt  y 

2M.Z'dx'-X'dz'=  C", 


1M 


dt 

yf  d  zf  —  zf  d  y' 
dt 


=.-  C". 


•    •    »*   • 


(249) 


18-1  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

But  §  190,  x' dy' '  —  y' d  x\  is  twice  the  differential  of  the  area  swept 
over  by  the  projection  of  the  radius  vector  of  the  body  M,  on  the 
co-ordinate  plane  x'  y\  and  the  same  of  the  similar  expressions  in 
the  other  equations,  in  reference  to  the  other  co-ordinate  planes; 
whence,  denoting  by  Az ,  Ay,  Axi  double  the  areas  described  in  any 
interval  of  time,  t,  by  the  projections  of  the  radius  vector  of  the  body 
J/",  on  the  co-ordinate  planes,  x'  y\  x'  z',  and  y'  z' ,  and  adopting 
similar  notations  for  the  other  bodies,  we  have 

dt  ' 

dt  .         . 

r 

dt 

in  which  C",  C",  C",  denote  the  sums  of  the  products  obtained  by 
multiplying  each  mass  into  twice  the  area  swept  over  in  a  unit  of  time 
by  the  projection  of  its  radius  vector  on  the  planes  x'  y\  %'  z\  y '  z' ';  and 
by  integrating  between  the  limits  tt  and  t\  giving  an   interval  equal   lo  I. 

2M.AZ=  C'.t; 
2M.Ay=  C"  f; 
2M.AX=  C'"t\ 

whence  we  find  that  when  a  system  is  in  motion  and  is  only  sub 
jected  to  the  attractions  or  repulsions  of  its  several  elements  upon 
each  other,  the  sum  of  the  products  arising  from  multiplying  the 
mass  of  each  element  by  the  projection,  on  any  plane,  of  the  area 
swept  over  by  the  radius  vector  of  this  element,  measured  from 
the  centre  of  inertia  of  the  entire  system,  varies  as  the  time  of  the 
motion.     This  is  called    the  principle    of  the    conservation  of  areas. 

It  is  important  to  remark  that  the  same  conclusions  would  be  true 
if  the  bodies  had  been  subjected  to  forces  directed  toward  a  fixed 
point.  For,  this  point  being  assumed  as  the  origin  of  co-ordinates, 
the  equation  of  the  direction  of  any  one  force,  say  that  acting  upon 
J/,  will  be,  §185, 

Yx  —  Xy  z=z  0 ; 


MECHANICS    OF    SOLIDS.  185 

and  the  second  members  of  Equations  (240)  will  reduce  to  zero; 
and  the  form  of  these  equations  being  the  same  as  Equations  (245), 
they  will  give,  by  integration,  the  same  consequences. 

INVARIABLE     PLANE. 

8  1S6. — Denoting  the  angles  which  the  resultant   axis  of  rotation 
makes  with  the  axes  x',  y',  z\  by  0X,  6V,  0Z,  we  have, 


iin 


_Av1_Ni_C_ 
C       *  -   K  ~  K  -   K' 

_Bv1J_Mi_C^ 
C0S  U*  -   K  ~  K  -  W 

Cvz        L         C" 


>  .    .    .    .     (250). 


These  constant  values  determine  the  position  of  the  resultant  or 
invariable  axis.  The  plane  at  right  angles  to  this  axis  is  called  the 
principal  plane.  The  position  of  this  plane  is  invariable,  and  it  is 
therefore  called  the  invariable  plane,  either  when  the  only  forces  of 
the  system  are  those  arising  from  the  mutual  actions  and  reactions 
of  the  bodies  upon  each  other,  or  when  the  forces  are  all  directed 
towards  a  fixed  centre. 

CONSERVATION     OF     KINETIC     ENERGY. 

§  187. — If,  during  the  motion,  two  or  more  bodies  of  the  system 
impinge  against  each  other  so  as  to  produce  a  sudden  change  in  their 
velocities,  the  kinetic  energy  of  the  system  will  undergo  a  change.  To 
estimate  this  change,  let  A,  B,  C  be  the  velocities  of  the  mass  my 
in  the  direction  of  the  axes  before  the  impact,  and  a,  ft,  c  what  these 
velocities  become  at  the  instant  of  nearest  approach  of  the  centres  of 
inertia  of  the  impinging  masses,  then  will 

A  —  «,     B  —  b,     C  —  c, 
be  the  components  of  the  velocities  lost  or  gained  by  m  at  the  instant 
corresponding  to  this  state  of  the  impact,  and 

m (A  —  «),     m  (B  —  ft),     m(C  —  c), 
the  components  of   the  forces  lost   or  gained.     The  same  expressions, 
with    accents,    will    represent    the    components    of    the    forces    lost    or 


186      ELEMENTS    OF   ANALYTICAL    MECHANICS. 

gained  by  the  other  impinging  bodies  of  the  system.     These,  by  the 
principle  of  D'Alembert,  §  71,  arc  in  equilibrio,  whence 

2  m  ( A  —  a )  6x  +  2  m  (B  —  b)  6y  -f  %  m  (C  —  c)6z  =  0. 

The    indefinitely  small    displacements    6x,  dy,  dz,  <fcc,  must   be   made 

consistently  with  the  connection  by  virtue  of  which  the  velocities  are 

lost  or  gained;    but  as  a,  b,  c  denote    the    components   of  the  actual 

velocities  of  the  body  whose  mass  is  m,  at  the  instant  of   its  nearest 

approach  to  that  with  which  it  collides,  this  condition  will  be  fulfilled 

if  we  make 

dx  =  a .  6t ;         dy  =  h  ,6t;         dz  as  c  .  6t. 

These  values   being   substituted  in   the   above   equation,  we   have, 

after  dividing  by  dt, 

Ilm(A  —  a)a  +  Zm(B  —  b)b  +  2  m  (C  —  c)  c  =  0  .    .  (251) 
or, 

2  m  (Aa  +  Bb+  Cc)  —  Z  m  (a2  +  &  +  c*)  =  0  .   .    .  (252) 

But  we  have  the  identical  equation, 

(        A*  +  B*  +  C2  -f  a2  +  #> 

(J  _  a)2  +  (B  -  by  +  (C-  C)2  =  j 

or, 


Aa  +  Bb  +  Cc  =   H 


-f-  c2  _  2  (^a  +  Bb  +  Cc). 

^l2  +  B*  +  (72       a2  +  fc2  +  g 2 

2  +  2~ 

(yl  _  a)9  _j_  (2?  _  5)2  +  ((7  _  c)2 


2 

which  in  Equation  (252)  gives, 

2m(J2  +  £2+<^)— 2m(a2  +  &2  +  c2)  =  2m[(.4  —  af^(B—bf+(C—  e)2], 
and  making 

^2  +  £2  +   C2  =  F2, 
«2    _j_   £2   _|_    c2    — -  w2? 

2wF2  —  2mw2  —  2m[(J  — a)2  +  (£  —  6)2  +  (C  — c)2]  ...    .  (253) 

whence  we  conclude,  that  the  difference  of  the  system's  kinetic  energy 
before  the  collision,  and  at  the  instant  of  greatest  compression,  is  equal 
to  the  kinetic  energy  which  the  system  would  have,  if  the  masses  moved 
with  the  velocities  lost  and  gained  at  this  stage  of  the  collision. 

Since  all  the  terms  of  the  preceding  equation  are  essentially  posi- 
tive, it  follows  that  at  the  instant  of  nearest  approach  of  the  impinging 
bodies  there  is  a  loss  of  kinetic  energy. 

If  the  impinging  masses  now  react   upon  each    other  in  a  way  to 


(a  —  Ay  +  (b  —  B'Y  +  (e—  C'f  =  i 


MECHANICS    OF    SOLIDS.  187 

cause  them  to  be  thrown  asunder,  and  A',  B',  C,  &c,  denote  the 
components  of  the  actual  velocities,  in  the  direction  of  the  axes,  at 
the  instant  of  separation,  then  will  the  components  of  the  velocities 
lost  and  gained  while  the  separation  is  taking  place,  be 

a  —  A',     b  —  B',     c  —  C\     &c,  <fcc. ; 
and  Equation   (251)   will  become 

2  m  (a  — A')  a  +  2,  m  (b  —  B)  b  +  2  m  (c  —  C')c  =  0, 
or, 

2  m  (a2  +  62  +  c2)  —  2m  (Aa  +  ^'6  +  C'c)  —  0  ; 

and  eliminating  Act  -f  i?'&  4-  C'c,  by  means  of  the  identical  equation, 

a*  %  &  +  c2  +  A'*  +  ^  2 
+  <7'2  —  2  (^4'a  +  B'b  +  (To), 

we  obtain, 

(«  — ^l')2 

2  m  (a2  +  ft2  +  c2)  —  2  m  (A'*  +  B'*  +  C"2)  =  —  2  » J    +  (b—B'f 

l  +  (c-C'Y 
and  making 

^'2  +  £'2  +  c"2  =  F'2, 

2mw2— 2  m  F'2=— 2m  [(«—^')2+ (6  — ^')2+(c—^r')2]     .     •     (254) 

All  the  terms  of  this  equation  being  essentially  positive,  it  follows, 
from  the  sign  of  the  second  member,  that  during  the  reaction  of  the 
bodies  by  which  they  are  separated  there  is  a  gain  of  kinetic  energy. 

If  the  loss  and  gain  of  velocities  after  be  the  same  as  before  the 
instant  of  greatest  compression,  then  will  there  be  no  loss  or  gain  of 
kinetic  energy  by  the  collision. 

PRINCIPLE     OF     LEAST     ACTION. 

§  188. — Resuming  Equation  (121),  and  employing  d  to  denote 
the  symbol  of  variation  in  the  Calculus  of  Variations,  we  may  write, 

1 6MV*  =  X6x  +  Y6y  4-  Z6z, 
and  substituting  for  X,  Y,  and  Z  their  values  in  Equations  (120), 

$6M V*  =3  Jf.i(tfte.d*  4-  dhj.dy  4-  cPz.dz). 

But  for  cPx . dx  4-  d2y .dy  4-  cPz.dz 

we  may  write 

d  (dx  .dx  4-  dy  .dy  4-  dz  .  6z)  —  (dx .  d  6x  4-  dy  .d  6y  +  dz .  d  dz), 


188      ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and  for  the  last  term, 

dx.ddx  +  dy.d6y  -f  dz .  d dz  =  \ .  6  (dx*  +  dy*  -f  dz2)  —  ^dds2; 
whence, 

6M  V*  =  M .  1  d  (pt .  6x  +  % .  d>  +  * .  dz); 
eZ*      V*  dt      '   ^  dt       r 


or, 


dMV.ds  =  M.dl-.dx  +dl.6y  +  d^.  6z\ 

\dt  r  dt      J  ^  dt       r 

and  by  integration, 

tfMV.d*  =  M^t.6,  +  d-l.Sy +  %*.). 

And  for  any  number  of  bodies, 

<52  f  MV .ds  =  Z  m(~  .6x  +(ll.6y  +  d4 .6z\ 
J  \dt  dt      u        dt      J 

Now  MV  is  the  body's  quantity  of  motion  and  is  the  measure  of 

the  intensity  of  the  force  that  produces  it.     MV.ds  is  the  elementary 

quantity   of   work    or  of   action ;    and    integrated   between    limits    will 

give  the  quantity  of  action  between  those  limits.     But  at  these  given 

limits 

6x  =  0  ;         6y  =  0  ;         dz  =  0, 

and  dJ2MV.ds  =  0. 

That  is  to  say,  in  the  motion  of  a  system  of  bodies,  the  curves  they 
describe  and  their  velocities  are  such  as  to  make  the  sum  of  the 
quantities  of  action  between  given  limits  on  their  respective  paths 
generally  a  maximum  or  minimum  ;  and  because  it  is  always  possible 
to  assign  to  each  body  a  definite  path  longer  than  any  assumed  for 
it,  the  quantity  of  action  is  obviously  a  minimum.  This  is  called 
the  principle  of  Least  Action. 

Because     ds  =  Vdt,     we  find  that     2  /  MV2dt     is  a  minimum,  or 

that  the  quantity  of  kinetic  energy  expended  during  any  given  time 
is  a  minimum. 

If  there  be  but  one  body  and  that  moving  upon  a  surface,  V  will 

be  constant  and     /  6s  will  be  a. minimum,  and  the  body  will  describe 

the  shortest  distance  between  any  two  points  arbitrarily  taken  on  its 
path.    . 


MECHANICS    OF    SOLIDS.  189 

§  189. — To  return  to  the  rotary  motion  of  a  single  body. 
If  for  the  impulsion  measured  by  Mv  \vc  take  the  moments  with 
reference  to  the  axes  x\  y\  z',  then,  since 

^rdx  *rdy  .rclz' 

if—-,         if-f-,         M  — 
dt  '  dt '  dt 

are  the  components  of  Mv,  it  is  clear  that  Equations  (249)  are  ex- 
pressions for  these  moments.  Designating  them  by  Z;,  Mt,  Ni%. 
respectively,  for  the  axes  z\  y\  x',  Equations  (229)  and  (249)  give 

Lt  =C    =  Cvzy 

M/  =  C"  =  BVyy 

Nt  =  C"  =  Av£; 

or,  squaring  and  adding, 

£2  + ^2  +  ^2  =  R 

C"2+  C"2+  C""2=F, 

A*vs*  +  B2vJ>  +  C*vt*  =  *» ; 

which  are  other  expressions  for  the  law  of  conservation  of  areas.  And 
it  is  evident  that  the  resultant  area,  or  moment  &,  is  a  constant 
quantity. 

It  has  been  shown  by  Poinsot  that  the  principal  plane  coincides 
with  that  diametral  plane  of  the  central  ellipsoid  of  inertia  which  is 
conjugate  to  the  instantaneous  axis. 

To  prove  which,  as  the  point  x' ,  y\  zr,  is  upon  the  instantaneous 
axis,  Equations  (193)  give, 


x        v        z 

j • 


(«) 


and,  as  it  is  also  upon  the  ellipsoid,  we  have, 

Ax'*  +  By'*  +  Cz"2  =  1. 
The    equation    of    the   tangent   plane    to    which,    through    the    paint 


x\  y',  z',  is 


Ax'x  -f  By 'y  +  Cz'z  =  1. 


190      ELEMENTS    OF    ANALYTICAL    MECHANICS. 

And,  therefore,  the  conjugate  diametral  plane  is 

Ax'x  +  By'y  +  [Cz'z  =  0; 
which  the  ratios  («)  transform  into 

Avxx  +  Bvyy  +  Cvzz  =  0. 

A    perpendicular    to    this    diametral    plane    makes  with    the   axes 
(.x\  y\  z',  angles  whose  cosines  are 

Avx  Bvy  Cvz 

k  '  k   !  k  » 

and,  therefore,  the  plane  coincides  with  that  of  resultant  rotation 
around  the  principal  axis,  but  generally  not  around  the  instantaneous 
axis ;  which  continually  shifts  its  position  both  in  the  body  and  in 
space,  and  coincides  with  the  principal  axis  only  when  rotation  takes 
place  about  one  of  the  natural  or  principal  axes  of  the  body. 

To  find  the  angle  made  by  the  instantaneous  with  the  axis  of  the 
principal  plane,  denote  it  by  0;  then  Equations  (194)  and  (196) 
give 

kvt  cos  0  =r  AvJ*  +  Bvy2  +  Cvz2 ; 

the  second  member  of  which  equation  denotes  double  the  kinetic 
energy. 

To  prove  this,  Equation  (210)  gives  for  the  moment  of  inertia, 

• 

2  mr2  Bs  A  cos2  a  -f  B  cos2/3  -f-  0  cos2y ; 
and  this,  multiplied  by  v£,  becomes, 

2  mv2  =  v2  2  mr2  =  Av*  +  Bv2  -f-  Cv}. 

,  Hence,  we  have, 

Xmv2 
vlCos9  =  — £-; 

which  is  constant  when  there  are  no  external  disturbing  forces.     This 
result    shows,    §  161,    that   under   such    circumstances   the  component. 
'  angular  velocity  with    reference    to   the    invariable  axis  and  invariable 
plane  is  constant. 

Let  6  denote  the  semidiameter  of   the  central  ellipsoid  coincident 


MECHANICS    OF    SOLIDS.  191 

with  the  instantaneous  axis,  and  e2  double  the  kinetic  energy,  or  simi 
of  the  living  forces;    then, 

2  mv*  =  vf  2  mr2  =  e2 ; 

and  this  gives,  because  of  equation  of  tangent  plane  at  end  of  semi- 
diameter, 

vt  =  6e. 

.which  shows   that   the   angular  velocity  is   proportional   to  that  semi- 
diameter. 

Whenever  the  moving   body  is  not   acted   upon  by  disturbing  ex- 
ternal forces,  Equations  (202)  give, 

Ad-£=vvv,(C-B), 
■£d£=vzvz(A-C), 

Multiplying  these  respectively  first  by    Avx,    Bvy,    Cvz,    and  then  by 
ran  vy>  v zi  we  obtain  by  addition, 

A*  vx  dvx  +  B*  vy  dvy  +  C'2  v%  dK  _  o, 
Avxdvx   +  Bvydvy  +  Cvzdvz    =  0; 
and  bv  integration, 

A2  v*  +  E*  v*  +  C2  v*  =  k% 
Av£  +  Bv*  -f  Cv?   =  <?2; 

which  values  are  constant.     And  thus,  as  equations  of   condition,  the 
laws  of  conservation  of  areas  and  of  kinetic  energy  are  ao-ain  found. 

Let  us  now  consider  what,  under  such  conditions,  will  be  the 
motion  of  the  central  ellipsoid,  whose  principal  axes  coincide  with 
those  of  the  moving  body. 

Referring  it  to  those  axes,  and  omitting  accents,  the  equation  of 
its  tangent  plane  through  a  point  x\  y\  z ',  where  an  instantaneous 
axis  cuts  the  surface  of  the  ellipsoid,  will  be 

Axx  -f  Byy'  +  Czz'  =  1  ; 


192      ELEMENTS    OF    ANALYTICAL    MECHANICS, 
which  the  relations 

x'  —  y'  ~  z'   ~  6  ~   ' 

of  §  159  transform  into 

Avxx  -f  Bvyy  -f  Cvzz  =  e. 

For  the  conjugate  diametral  plane  parallel  to  this  tangent  plane, 
we  have  the  equation, 

Avxx  -f  Bvyy  -f  Cvzz  =  0. 

A  common  perpendicular  to  these  two  planes,  drawn  from  the 
centre,  coincides  with  the  resultant  axis  or  that  of  the  invariable  plane, 
§  186;  its  direction  cosines  are 

Avx  _  Bv„  -  Cvz 

-—.  =  cos  Vx,        —  2  =  cos  6yy        —  =  cos  0X ; 

and  its  length  is 

e 


e 


which  is  constant. 


The  central  ellipsoid  will,  therefore,  roll  upon  the  tangent  plane, 
whose  distance  from  the  centre  is  constant,  preserving  contact  at  con- 
secutive points,  xr,  y\  z\  the  poles  where  the  instantaneous  axis  cuts 
the  ellipsoid. 

A  principal  axis  describes,  therefore,  a  conical  surface  around 
the  axis  of  the  invariable  plane,  while  the  lines  within  the  body  that 
successively  become  the  instantaneous  axis  form  another  conical  sur- 
face about  the  principal  axis.  The  elements  of  these  cones  having  a 
common  point  at  the  centre  of  inertia,  are  always  tangent  to  one 
another  along  a  common  element,  and  which  common  element  is,  for 
the  time  being,  the  instantaneous  axis.  So  that  the  question  of  a 
body's  rotary  motion  is  reduced  to  that  of  one  cone  rolling  over  the 
surface  of  another  having  the  same  vertex. 

The  initial  conditions  of  the  motion,  the  nature  of  the  forces,  and 
their  mode  of  action  adjust  the  angles  of   the  rolling  and  the  direct- 


MECHANICS    OF    SOLIDS, 


193 


ing  cone ;  and  these  angles  determine  the  number  of  revolutions  which 
the  instantaneous  must  make  about  the  principal  axis  of  the  body, 
to  carry  this  latter  once  around  the  invariable  axis. 

How  it  is  that  an  instantaneous  axis  may  be  at  rest,  in  rolling 
contact,  is  clear  from  the  fact  that  a  point  in  the  body  first  approaches 
and  then  recedes  from  contact;  thus  reversing  the  direction  of  its 
velocity  with  reference  to  the  tangent  plane.  That  velocity  must, 
therefore,  change  its  algebraic  sign  and  become  zero  at  the  instant  of 
contact. 


§  190. — The  free  motion  of  a  body  about  a  centre  being  that  of 
one  cone  rolling  upon  another,  the  problem  is  determinate  when  the 
cones  and  their  velocity  of  description  are  known. 

For  the  rolling  cone,  fixed  in  the  body,  the  relations  of  §  159 
give 


x 


vn 


cos  a,  =  -=  =  — , 


v 


i 


coSi3;  =  ^  =  ^, 


cosy; 


z 
6 


v. 


V, 


V, 


These  are  functions  of  the  time,  by 
eliminating  which  the  equation  of 
the  cone  is  obtained. 

To  find  the  directing  cone  de- 
scribed by  the  instantaneous  around 
the  invariable  axis :  if  those  axes 
lie  in  the  same  plane  with  that  of 
the  rolling  cone,  it  is  clear  that  the 
directing  cone  will  be  determined 
when,  for  the  point  /  on  the  unit 
sphere,  we  know  the  radii  of  curva- 
ture, 

r  =z  sin  «,         r'  =  sin  j3, 


and  o)  the  angular  velocity  with  which 


vi 


194      ELEMENTS    OF    ANALYTICAL    MECHANICS. 

the  plane    of   the    axes  revolves   about   OA,  the    invariable   axis ;    and 
which  is  generally  called  the  precessional  velocity. 

Now,  when  the  elementary  curve,  ds,  of  the  rolling  cone  is  applied 
externally  to  its  equal  correlative  element  ds'  upon  the  directing 
cone,  it  is  evident  that  the  corresponding  angular  motion  around  the 
instantaneous  axis  must  be  the  sum  of  the  two  angles  subtended  in 
their  respective  circles  by  these  equal  arcs,  when  the  rolling  is  ex- 
ternal to  the  directing  cone,  which  is  supposed,  or  that 


ds  /l 


v  ■  —  — 
l~  dt 


\a        a  J 


This  motion  takes  place  in  the  plane  AIB  normal  to   01;   but 


rji  if  2 


1  —  r2 '  1  —  r'2 


'<>' 


and,  therefore, 


ds(\/l  —  r*       yl 


r 


2^ 


h 


or, 


(1)     .     . 


V>  =  d\— —-  +  -      r' 

ds  rr' 

dt'      lr  \/\~?*  +  r'  ^/T^r* 


which  equation  gives  the  value  of  r  in  terms  of  v{J  s,  and  r'\  and, 
therefore,  when  these  are  known  in  terms  of  t,  it  determines  the 
directing  cone. 


Again,  putting 


ds  .   , 

—  —  r(D  =  r  a) , 
dt  ' 


we  have 

(2) <*>  sin  a  =  (*)'  sin  j3, 

and 

v{  =  0)  cos  a  -{-  <*)'  cos  /3. 

Also,  the  motion  of  the  point   C  on  the  rolling  axis  gives, 

(3)  .     .     .     .     .     vt  sin  (3  =  (0  sin  (a  -f-  (3). 

Hence, 

o) '\  4»r.  ; vt : :  sin  /3  :  sin  a  :  sin  (a  +  (3)  ; 

the  geometric  construction  of   which   result   is,  evidently,  a   parallelo- 
gram of  rotations  w,  vp  (*>'.  described  on  the  axes  A,  I7  B. 


MECHANICS    OF    SOLIDS 


11*5 


From  the  relations  just  found  we  readily  get 

ds  sin  a  .  sin  3 

.     .     .     .      —  —  v.   - 
dt 


(4) 


-tl 


£  '   sin  («  +  /?)' 
with  which  equation  (1)  is  identical. 

If  the  movable  cone  roll  externally  upon  the  directing  cone,  all 
the  angular  velocities  are  similar — 
either  all  positive,  or  all  negative. 
Of  such  motion,  the  common  top 
spinning  around  its  point  as  a  fixed 
centre,  while  its  axis  gyrates  slowly 
in  precessional  revolution,  furnishes 
a  familiar  instance. 

But  when  the  movable  cone  rolls 
internally  upon  the  surface  of  the 
directing  cone,  then  w  is  in  direction 
reverse  to  vf  and  (*)',  it  being  posi- 
tive if  they  are  negative,  and  nega- 
tive when  they  are  positive.  In  this 
case  the  angle  j3  is  negative,  and 
Equation  (3)  becomes, 

vt  sin  (3  =  —  g)  sin  (a  —  |3). 

The  precessional  revolution   of   the  earth's  axis,  B,  around  A,  the 
axis  of   the  ecliptic,  is   an    interesting   example   of   this   second   case. 
The  rotation  of  the  earth  about  its  own  and  about  the  instantaneous 
axis   being   direct    (negative),    that 
of   the    precession    must    be    retro- 
grade,   or   positive.     Its    period    is 
25868  years;   the  obliquity  of   the 
ecliptic    is   23°  27'  30";    and    the 
length  of   the  circle  traced   by  the 
instantaneous   axis    on    the    surface 
of    the    earth    is,    therefore,    about 
52,240,000  feet.     From    these  data 
it  is  easily  calculated  that  the  roll- 
ing cone  goes  five  and  a  half  feet 
per  day,  and  that  the  radius  of  its  base  is  0.88  feet  only. 


-G$ 


196       ELEMENTS    OF    ANALYTICAL    MECHANICS. 

§  191. — To  find  the  cone  described   in  the  body  by  the  instanta- 
neous axis  when  there  are  no  external  forces,  the  relations, 

vx  =  ex,         vy  =  ey,         vz  —  ez, 

transform  the  law  of  areas  into 

^2^2  +  £2pfy2  +  cYe2  =  1. 

Also,  for  the  central  ellipsoid, 

Ax*  -f  By*  -f  Cz2  =  1 ; 


if  we  put 


we  have, 


A-.l       b-I        a-1 

a*  b*  c* 


#2        y2       02 

^  +  T2  +  S  =  1# 


And  from  these  expressions  we  obtain, 

^2  (a2  _  ^2)  22  +  £2  (J2  _^2)  y2  +  <?2  (c2  ^.^  *2  =  Q, 

the  required  equation  of  the  cone. 

For  the    cone    described    in    the    moving   body    by    the    invariable 

axis,  let  x,  y,  z  denote  £  point  upon  that  axis  at  the  distance  p  from 
the  centre,  then  its  direction  cosines  are 

p  k     '         jt>  A:  jo"       &    ' 

and  these  give 

a*  +  y2  _J_  22  _  j0f 

the  equation  of  a  sphere,  with  the  same  tangent  plane  as  the  ellipsoid. 
Substituting  these  values  in  the  equation  of  living  force, 

Avs*+  Bv*  +  Cv?=ze\ 
and  reducing,  we  get, 

ofo*  4.  #ty2  ^  C2Z2  —  pi ; 


MECHANICS    OF    SOLIDS 


197 


and,  therefore, 

(a3  _  p2)  Z2  +  (fc2  _  j0)  y2  +  (e2  _^2)  s2  _  Q, 

is  the  equation  of  the  cone. 

The  equations  found  show  that  the  cones  are  elliptical,  and  that 
their  axes  of  symmetry  coincide  with  the  principal  axes  of  the  body 
and  of  its  ellipsoid  of  inertia.  They  are,  therefore,  right  cones,  whose 
bases  are  ellipses;  the  equations  for  which  are  found  by  making  x 
or  z  constant. 

If   a,    b,   c   be   unequal   and    denote   the  semiaxes   of   the   central 
ellipsoid  in  the  order  of  their  relative  length,  then  the  cones  are  de- 
scribed about  x,  the  greater  axis,   when  b  is  less  than  p}  and  around 
z,  the  least  axis  of  the  ellipsoid,  or 
body,  when  b  is  greater  than  p.    But 
if  c  be  not   less   than  p,  the  cones 
are  imaginary. 

Let  secant   planes   cut   the  axes 
x,  y,  z,  so  as  to  form  a  cube  at  th 
centre,  and  assume,  successively,  for 
the  constant  p    different   increasing 
values,  all  greater  than  c;  then  the 

elliptical  bases  of  the  cones  will  each  increase  in  size  and  eccentricity 
until  p  is  taken  equal  to  b,  when  the  corresponding  cone  opens  out 
into  two  asymptotic  planes  intersecting  in  the  axis  y,  and  whose  traces 
in  the  plane  xz  are,  for  the  instantaneous  axis, 


A  .  /a 

=  ±*7tV: 


JO2 


C      p2  —  c2 


»     » 


and  for  the  invariable  axis, 


=±^i 


—  JO2 


JO2 


,2* 


Beyond  these  limiting  planes,  if  we  give  p  still  greater  values,  the 
body  spins  around  x}  and  its  cones  for  x  give  hyperbolas  with  a 
secant  plane  perpendicular  to  x.  Each  value  of  p  has  its  particular 
cone,  either  about  x,  or  about  z;  and  the  limiting  planes  divide  the 
space  around  y  into  regions,  one  for  rotation  around  xy  the  other  for 


198       ELEMENTS    OF    ANALYTICAL    MECHANICS. 

rotation  around  z.  The  figure  also  shows  that  sections  of  the  cones 
by  planes  perpendicular  to  y  are  all  hyperbolas.  And  if  we  imagine 
a  sphere,  instead  of  the  cube,  at  the  centre,  its  traces  with  the  cones 
will  be  spherical  ellipses. 

For  symmetrical  solids  the  cones  become  circular  and  the  ellipsoid 
one  of  revolution,  around  z  if  a  be  equal  to  b,  around  x  if  b 
equal  c.     And  if 

A  =  B  =  C, 

the  ellipsoid  is  .a  sphere,  with  permanent  rotation  for  any  diam- 
eter. 

The  only  condition  for  rotation  about  a  permanent  axis  has  been 
shown,  §  178,  to  be  that  the  body  must  revolve  about  one  of  its 
principal  axes.  The  rolling  and  the  fixed  cone  then  reduce  to  their 
axes,  and  the  invariable,  instantaneous,  and  rolling  axes  coalesce  into 
a  single  line,  or  axis,  normal  to  the  tangent  and  invariable  planes. 
That  only  a  principal  axis  can  be  permanent  is  clear,  for  a  diameter 
is  normal  to  the  tangent  plane  of  an  ellipsoid  only  at  the  ends 
of  its  principal  axes. 


PLANETARY     MOTIONS. 

§  192. — When  the  only  forces  are  those  arising  from  the  mutual 
attractions  of  the  several  bodies  of  the  svstem  for  one  another,  the 
second  members  of  Equations  (239)  reduce,  as  we  have  seen,  §  183, 
to  zero,  and  those  equations  become 

MM        <PX 


(255) 


Let  us  now  find  the  motion  of  any  one  body  of  the  system  in  refer- 
ence to  any  other,  taken  at  pleasure.  This  latter  body  will  be  called 
the  central,  the  former  the  primary,  and    the  others,  collectively,  the 


MECHANICS     OF    SOLIDS.  199 

{tertarbatiuy  bodies.     Let  the  central  and  primary  bodies  be  those  whose 

masses  are  M  and  M,  respectively;    the  perturbating  bodies  those  whose 

masses   are  M '  ,  M    ,  &c.      The   first   of   the   above   equations   mav   be 

written 

__   d*  x        ._    d*  x  __      d^xjt 

M-if+M'-i¥  +  1M»-dii  =  ('  ■  •  •  c-30' 

If  the  perturbating  bodies  alone  acted  upon  one  another,  the  last  term 
would  be  zero ;  and  when  the  action  of  the  central  and  primary  are 
included,  the  numerical  value  of  this  term  will  result  from  the  action 
of  these  latter  bodies.  Denote  the  reciprocal  action  of  any  two  bodies 
upon  one  another  by  writing  their  masses  within  the  parenthetic  sign, 
and  use  the  subscript  x  to  denote  the  component  of  this  action  parallel 
to  the  axis  x.     Then  will 

S  (M M,,).  +  £  {M,  M„  ).  -  2  Mu  $£f  =  0  ; 

adding  this  to  the  next  equation  above,  we  get 

Taking  the  movable  origin  at  the  centre  of  the  body  M.  we  have 

xt  ==  x  —  x\  and  d*  xt  —  d2  x  —  d*  x\ 
which,  substituted  above,  gives 

dividing  by  M  +  Mt  and  multiplying  by  M,  there  will  result 
,,  d% x      M.M,    d*x'  M        _  ,  __  __  ,  M        -,'/»'«'•••» 

The  value  of   the    first    term    results    from    the  eomponent  action   of 
the  primary  and  perturbating  bodies  upon  M\    whence 

M.  ~  -  [ (MMt),  -  S  (MM„U  .  0  i 

from   which  subtracting  the  equation  above,  there  will  result 
M  M      d*  r'  M  M 


200 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


Dividing  by  the  coefficient  of  the  first  term,  and  treating  the  other  two 
of  Equations  (255)  in  the  same  way,  we  finally  get 


d*z'     M+Mt 

U~  mTm 


.(MM,),+  -.Z(MMJ 


M 


M. 


.X(MfMtX  =  0, 


}   (258) 


Which,  by   integration,   will  give  all  the  circumstances  of  the  primary's 
motion  in  reference  to  the  central   bod  v. 


LAWS    OF    CENTRAL    FORCES. 


§  193. — A  central  force  is  one  which  is  directed  towards  a  centre, 
movable  or  fixed,  and  of  which  th.e  intensity  is  a  function  of  the  dis- 
tance from  the  centre.     The  forces  of  nature  are  of  this  description. 

If  the  perturbating  bodies  did  not  exist,  then  would  the  action  oi. 
the  primary  be  directed  to  the  central  body  as  a  centre,  the  Equations 
(258)  would  reduce  to  their  first  two  terras,  and,  denoting  the  distance 
from  the  central  to  the  primary  by  r',  the^y  would  be  written, 


d*x       M+Mt  M+Mt  x' 

-;-,-  =  tf—vt  •  {MM ),  =  - --— —  .  (MM) .  -  ; 
dt*       M.M     v         '"       M.M.    v  •'    r" 


rfV      M+M 


dt* 


M.M 


'-\MM\, 


M+M 


!/ 


M.M 


'-.(MM,).J-t; 


\. 


d*z'      M+M,    ,,_   „        M+M,    #M<        z 

-ft  =  ,flf  •  (M.MX  =  -rr-rr  .  (MM) .  -,. 
dt*       M.M     v  '         M.M.     v         '     r' 


(250) 


Multiply  the  first  by  y',  the  second  by  x',  and  take  the  difference  of  the 
p  jducts ;  also  multiply  the  first  by  z',  the  third  by  x\  and  take  the  differ- 
ence of  the  products;  and  again  the  second  by  z',  the  third  by  y\  and  take 
the  difference  of  the  products  :    there  will  result,  omitting  the  accents, 


d'y 


d*x 


de 

,  x 

de 

y 

=  u. 

(Px 

dt> 

.  Z 

d'z 

,x 

=  0, 

<Pz 

IT* 

y 

<?y 
dt*' 

z 

=  0; 

MECHANICS    OF    SOLIDS. 


201 


which,  being  integrated,  give 


dy  dx 

—  .  x  —  — .  y  =  V , 
dt  dt    y  ' 

■^ —  •  Z  —  — —  .  X  =    K/    . 

dt'y      dt  °    ' 


(26C) 


in  which  C,  C",  and  Q"  are  the  constants  of  integration. 

Maltiplying  each   by  the  first  power  of  the  variable   which  it  does 
not  contain,  and  adding,  we  have 

C"2+  C"y+C///x=0; 

which  is  the  equation  of  an  invariable  plane  passing  through  the  cen- 
tre, and  of  which  the  position  depends  upon  the  constants  C\  C'\  C/". 
Whence  we  conclude  that  the  primary  deflected  by  the  central  body 
alone,  will  describe  a  plane  curve  of  which  the  plane  will  contain  the* 
centres  of  both. 

Take  the  co-ordinate  plane  xy  to  coincide  with  this  plane, 
and  the  Equations  (260)  will  reduce  to 


dt  d  t   y 

Transform  to  polar  co-ordinates;  for  this  purpose  we  have 

x  =:  r .  cos  a  ;    y  =  r .  sin  a  ; 

differentiating, 

d  x  =  dr  cos  a  —  r  sin  a  d  ar 

dy  =  dr  sin  a  +  r  cos  a  da. 
Substituting  in   Equation  (261),  we  find 

dy 


(261) 


dt 


dx  ji  da-       nr 

.x —  .y  =  r .  —  =  C 

dt    y  dt 


•     •     •     •     • 


(262) 


integrating  again,  we  have 

J*r\da=C't  +  C"f 
and  taking  between    the   limits  r#r  a.t  and  rilf  aif1  corresponding  to  tbe 


time  tt  and  *„, 


fr'a'  r\do.  =  C'(tit-tt) (263) 


202  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

But  f  r'2  d  a  is  double  the  area  described  by  the  motion  of  the  radius 

vector;  whence  we  see,  Equation  (263 ),  that  the  areas  described  by  the 
radius  vector  of  a  body  revolving  about  a  centre,  are  proportional  to  the 
intervals  of  time  required  to  describe  them. 

Making,  in  Equation  (263),  tit  —  tt  equal  to  unity,  the  first  member 
becomes  double  the  area  described  in  a  unit  of  time.  Denoting  this 
by  2  c,  that  equation  gives 

C"  =  2  c 

» 

Placing  this  in   Equation  (263),  we  find 


I  r  .a  a 

J    r..  a.. 


'..-'.  =  —  -f-e (26+) 

That  is  to  say,  any  interval   of  time  is  equal   to  the  area  described 
in  that  interval,  divided  by  the  area  described  in  the  unit  of  time. 

The  converse  is  also  true;  for,  differentiating  Equation  (262), 

we  find 

d*  y  d2  x 

77     "  d? 


t4  x  —  -tt  y  =  ° ; 


d^  v  d  x 

Multiplying  by  My  and  replacing  M.  j-J  and  M .  -j-j   by  their  vahi'n 

in   Equations  (120),  there  will  result 

Yx  —  Xy  =  0      .......     (265) 

which  is  the  Equation. of  the  line  of  direction  of  the  force;  and  having 
no  independent  term,  this  line  passes  through  the  centre.  Whence  we 
conclude,  that  a  body  whose  radius  vector  describes  about  any  point 
,*ieas  proportional  to  the  times,  is  acted  upon  by  a  force  of  which  the 
line  of  direction  passes  through  that  point  as  a  centre.  The  force  will 
be  attractive  or  repulsive  according  as  the  orbit  turns  its  concave  or 
convex  side  towards  the  centre. 

Replacing  C  by  its   value  2  c,  in   Equation  (262)^  and  divi 
ding  by  r\  we  have 

,  ,  da        2  c  . 

17°°  7    •   •   ' (266> 


MECHANICS    OF    SOLIDS.  203 

The  first  member  being  the  actual  velocity  of  a  point  on  the  radius 
vector  at  the  distance  unity  from  the  centre,  is  called  the  angular  ve- 
locity of  the  body.  The  angular  velocity  therefore  varies  iniersely  as 
the  square  of  the  radius  vector. 

Multiply  Equation  (26(5)  by  d  s,  and   it   may  be    put   undei 

the  *brm, 

d  s  2  c 


d  t  rda.1 

d  s 

but  — '- — ,  is  equal  to  the  sine  of  the  angle  which   the  element  of  the 

orbit  makes  with  the  radius  vector,  and  denoting  by  p  the  length  of 
the  perpendicular  from  the  centre  on  the  tangent  to  the  orbit  at  the 
place  of  the  body,  we  have 

r  .  d  a 
p  =  r.  — — , 
d  s 

and 

V=— (267) 

P 

whence,  the  actual  velocity  of  the  body  varies  inversely  as  the  distance 
of  the  tangent  to  the  orbit  at  the  body's  place,  from  the  centre. 

§  19  4. — Denoting  the  intensity  of  the  acceleration  on  Mt  by  F\  sub- 
stituting M \  .  F .dr  for  Xdx  +  Ydy  +  Z dz,  writing  Mt  for  M  in  the 
coefficient  of  V2  in  Equation  (121),  and  differentiating,  we  find 

VdV=  -Fdr\ 

and  taking  the  logarithms  of  both  members  of  Equation  (267), 

log  V  =  log  1c  —  log  p ; 

differentiating, 

dV dp 

~V~p1 

and  dividing  the  equation  above  by  this, 

n->:,.g.i!r.»J,.jr       .  .  .  (2«8) 

13 


204 


ELEMENTS  OF  ANALYTICAL  MECHANICS 


Whence  we  conclude  that,  the 
velocity  of  a  body  at  any  point 
of  its  orbit  is  the  same  as  that 
which  it  would  have  acquired  had 
it  fallen  freely  from  rest  at  that 
point  over  the  distance  ME,  equal 
to  one-fourth  of  the  chord  of  cur- 
vature M  G,  through  the  fixed  cen- 
tre— the  force  retaining  unchanged 
its  intensity  at  M. 


§  195. — Resuming  Equations  (120),  we  have 


d  x 

d2x  dt 

X=  M'-—  —  M — , 

dt1  dt 


and  performing  the  operation  indicated,  regarding  the  arc  of  the  orbit 
as  the  independent  variable,  we 'have,  after  dividing  both  numerator  and 
denominator  by  d  sz, 


dt  6} x      dx   d? t 


X=M. 


ds   d  a2       d  s    ds* 


ds2   d* x      dx   da?   d2  I 


DUt 


whence, 


dt' 

-'[ 

.dt'  ds* 

ds3 

ds*    d2t 

de'd**~ 

d2s  t 

~  d~i2 ; 

~    d2x 

dx   d2  s~i 

In  like  manner, 


™>[*#*-£l' 


d* 


d  z   d*  s' 


8qnanng  and  adding, 


MECHANICS    OF    SOLIDS. 


205 


- 1  &!*  &h  SS !  ■  *■ 


X'+Fa  +  Z2=:^ 


T™    d2  *  Idx   d%x      dy   d}y      dz   d?  z\    l|_ 


d  d       d  I    (/  s2 
/<*«■      dy'      dz\     '&•<* 

bnt,  denoting  the  radius  of  curvature  by  p,  we  have 

(£f\*     (d'yV     l<P*V      1* 

and  multiplying  the  second   term  of  the  second   member  of  the  prece- 
ding equation  by  -,  it  may  be  put  under  the  form, 

MV*  M.dUtdx      d*x      dy      <P p      dz      d%z' 


_£  .drsidx      d'1  x      dy      d*  y      dz       dl z\ 

~~dT~  \d~s  *p  17 + n ' p  i?  +  an' p dtp 


or, 


^d  s       d  s2      d  s    r  d  s2      d 

MV*    M.d's 

•  cos  o ; * 


dp 


in  which  d  denotes  the  angle  made  by  the  element  of  the  curve  and 
radius  of  curvature ;  also 

d  x1      d  ?/2      d  z* 

d  «»  T  d  s2  T  d  *2         r 

whence,  substituting  for  X*  -f-  Y*  -f-  Z*  its  value  B?r  we  have 

and  comparing  this  with   Equation  (56)  we  find  that  R  is  equal  to  the 
resultant  of  the  two  component  forces 

and  iff  •  -— ,t 

p  of  tiy 

which  make  with  each  other  the  angle  6.     But  6  is  equal  to  90°,  and 

therefore 

M'V* 


£*  = 


j?) (260> 


8e*  Appendix  No.  2. 


206  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

The  second  of  these  components  is,  Equation  (13),  the  intensity  of 
the  reaction  of  inertia  in  the  direction  of  the  tangent,  and  the  first  is 
therefore  its  reaction  in  the  direction  of  the  radius  of  curvature. 

This  first  component  is  called  the  centrifugal  force,  and  may  be  de- 
fined to  be  the  resistance  which  the  inertia  of  a  body  in  motion  oppose* 
to  whatever  deflect*  it  from  its  rectilinear  path.  It  is  measured,  Equa- 
tion (269),  by  the  living  force  of  the  body  divided  by  the  radius  ot 
curvature.  The  direction  of  its  action  is  from  the  centre  of  curvature, 
and  it  this  differs  from  the  force  which  acts  towards  a  centre,  and 
which   is  called  centripetal  force.     The  two  are  called  central  forces. 

If  the  component  in  the  direction  of  the  orbit  be  zero,  then  will 

and  denoting  the  centrifugal  force  by  Fn  we  have 

F<  =  ~y- (27°) 

and  integrating  the  next  to  the  last  equation,  we  have 

in  which  C  is  the  constant  of  integration.  Whence,  the  velocity  will 
be  constant,  and  we  conclude  that  a  body  in  motion  and  acted  upon 
by  a  force  whose  direction  is  always  normal  to  the  path  described,  will 
preserve  its  velocity  unchanged. 

These  laws,  except  that  expressed  by  Equation  (268),  are  wholly  in- 
dependent of  the  intensity  of  the  extraneous  force  and  of  the  law  of  its 
variation.     Not  so,  however,  of 


THE    ORBIT. 

§  196. — To  find  the  differential  equation  of  the  orbit,  multiply  the 
first  of  Equations  (259)  by  2  d  x,  the  second  by  2  d  y,  add  and  inte- 
grate ;   we  find,  omitting  the  accents, 

d*  +  dy>  _M+M,     f{MM.    2xdx  +  2ydy^ 
d?        ~   M.M,    'J\mm<>'  r 


MECHANICS    OF    SOLIDS.  207 

but 

r2  =  #s  -f-  y2,  and  rdr  =  xdx-\-ydy\ 

also 

x  =  r  cos  a  ;     y  s£  r  .  sin  a  ; 

d  x  =  —  r  sin  a  rf  a  -f-  cos  a  c?  r ; 

dy  =  r  cos  a  rf  a  -f-  sin  a  6?  r  ; 

and,  Equation     (266), 

12c 


d  t       r1  d  a 
These  substituted  above,  give 


Make 


H+^*<%£  ■***>*« 


u,  and  therefore    —  =  —  d  w, 


r  ra 


substitute  above,  differentiate  and  reduce,  there  will  result 

\r/V  '  J/.Jf        V  ''  L       M         T         if-      J1 

;     .     • 
and   making 

^    =*     I       i^      +       ^     ]  =  relative  acceleration  on  Mt   v   (271) 

V  ■■ 
From   which    the   equation    of  the   orbit   may  be   found  fry  "integration, 

when  the  law  of  the  force  is  known ;  or  the  law  of  the  force  Seduced, 

when  the  equation  of  the  orbit  is  given. 

In  the  first  case,  the  integral  will  contain  three   arbitrary  constants 

•  •  • 

— two  introduced  in  the  process  of  integration,  and  the  -third,  •,  c,  £exjst- 
iri£  in  the  differential  equation.  These  are  determined  ty  the  initial 
or  other  circumstances  of  the  motion,  viz. :  the  body's  velocity,  its  dis 
tance  from  the  centre,  and  direction  of  the  motion  at  a  given  instant.. 
The  general  integral  only  determines  the  nature  of  the  orbit  described  : 
the  circumstances  of  the  notion  at  any  given  time  determine  the  species 
and  dimensions  of  the  orbit. 


208  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

In  the  second  case,  find  the  second  differential  coefficient  of  u  in 
regard  to  a,  from  the  polar  equation  of  the  curve ;  substitute  this  in 
the  above  equation,  eliminating  a,  if  it  occur,  by  means  of  the  relation 
between  u  and  a,  and  the  result  will  be  F,  in  terms  of  u  alone. 


SYSTEM    OF    THE    WORLD. 

§  197. — The  most  remarkable  system  of  bodies  of  which  we  have 
Riiy  knowledge,  and  to  which  the  preceding  principles  have  a  direct 
application,  is  that  called  the  solar  system.  It  consists  of  the  Su?i, 
the  Planets,  of  which  the  earth  we  inhabit  is  one,  the  Satellites  of  the 
planets,  and  the  Comets.  These  bodies  are  of  great  dimensions,  are 
spheroidal  in  figure,  are  separated  by  distances  compared  to  which 
their  diameters  are  almost  insignificant,  and  the  mass  of  the  sun  is 
so  much  greater  than  that  of  the  sum  of  all  the  others,  as  to  bring 
the  common  centre  of  inertia  of  the  whole  within  the  boundary  of 
its  own  volume. 

These  bodies  revolve  about  their  respective  centres  of  inertia,  are 
over  shifting  their  relative  positions,  and  our  knowledge  of  them  is  the 
f  suit  of  computations  based  upon  data  derived  from  actual  observation 

Kepler  found  ; 

i;  That  the  areas  swept  over  by  the  radius  vector  of  each  planet 
about  the  sun,  in  the  same,  orbit,  are  proportional  to  the  times  of  de- 
scribing, them. 

II.  That  the  planets  move  in  ellipses,  each  having  one  of  its  foci  in 
the  sun's  centre. 

III.  That  the  squares  of  the  periodic  times  of  the  planets  about  the 
tun,  are  proportional  to  the  cubes  of  their  ?nean  distances  from  that 
body. 

^  These  are  called  the  laws  of  Kepler,  and  lead  directly  to  a  knowl- 
edge of  the  nature  of  the  forces  which  uphold  the  solar  system. 


CONSEQUENCES    OF    KEPLF.r's    LAWS. 

§  198. — The  first  law  shows,  §  193,  that  the  centripetal  forces  which 


MECHANICS    OF    SOLJDS.  209 

keep  the  planets  in   their  orbits,  are  all  directed    to   the    sun's   centre ; 
and  that  the  sun  is,  therefore,  the  centre  of  the  system. 

§  199. — What  law  of  the  force  will  cause  a  primary  to  describe 
about  a  central  body  an  ellipse  having  one  of  its  foci  at  the  centre  of 
the  latter  ?     The  equation  of  the  ellipse  referred  to  its  focus  as  a  pole  is 


r  = 


\  +  e  cos  a  ' 

whence, 

1  1  -h  e  cos  a 

r  =  U=  a  (1-7)'' 

and, 

<P  u        —  e  cos  a 
Jtf  =  a  (1  -  e1)' 

which,  substituted  in  Equation  (272),  give 


=  4c2  w2 


—  e  cos  a       1  4-  «  eos  av 
a  (1  -  ?)  +    a  (1  -  e2)  j  * 


reducing  and  replacing  w  by  its  value  -,  we  have 


•  4  c8  1  ' 

^  =  —r-. §\  *  ~i (273) 

a(l  —  r)    r 

and  from  which  we  conclude,  that  the  only  law  for  the  relative  accel- 
eration, is  that  of  the  inverse  square  of  the  distance. 

§  200. — Conversely,  let  the  force  vary  inversely  as  the  square  of  the 
distance;  required  the  orbit. 

Denote  by  kt  the  reciprocal  attraction  of  one  unit  of  mass  upon  an 
other  at  the  unit's  distance;    then  will 

[MM.)  =  M.Mr^; 

and,  Equation  (271), 

F=ki   (M+M4).u'  =  k,.m.u%; 

in  which 

m  =  M+Aft (273)' 


210.  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and.  Equation  (272), 

d*  u  k,.m 


-f-  u  = 


! 

X 

multiplying  by  2  d  u  and  integrating, 

d  u*       2  k,  .m 


da}  4  .  c2 

whence 


.  u  —  u*  +  C  ...     (274: 


—  d  u 

da  = 


2k,.m 


v/CrM.:^p..._iy 


the  negative  sign  being  taken,  because 


rf  /• 


da  da  •  r%i  d  a, 

'k, .  m\2 


.     .     (275) 


Place  under  the  radical   I  -j- r  I  —  |-f-r  I  i  and  we.  may  write. 


1  —  d  u 

a  a  = 


v(§F  /  EM 

and  integrating, 


k.  .m 

a  +  <p  =  cos 


</%&?*» 


in  'which  f  is  the  constant  of  integration. 

Replacing  u  by  its  value,  taking  cosine  of  both  members  and  solving 
with  respect  to  ?•,  there  will  result 

4  c8 


k, .  m 
r  = 


1+\/'1  +  (^)i-(7-cos("+?) 

which  is  the  equation  of  a  conic  section,  naving  its  pole  at  the  central 


MECHANICS    OF    SOLIDS.  211 

body.  To  find  the  precise  curve,  we  must  find  C.  To  do  thi&,  denote 
by  r4  the  initial  value  of  the  radius  vector,  and  by  ey  the  angle  which 
the  orbit  makes  with  rt  at  the  point  of  intersection  therewith.  Then, 
Equation  (275), 


du 

1 

• 

d  a 

T> 

tan  ey  ' 

and  this  in  Equation  (274)  gives 

c-     x 

2  h, .  m 

U    2           2 

r*  sin* 

e, 

4c*r,   ' 

but, 

Equation  (267), 

1 

v? 

V%r 

4         4 

r,2  sin*  e,       4  c2       4  ca.r. 


•     •     • 


(275)' 


in  which  Vt  is  the  velocity  corresponding  to  rt ;   hence, 

4cVy 
which,  substituted  in  the  equation  of  the  curve,  gives 

4  c2 


kJ .  m 

r  = 


hV^^-(^-^~<-+»> 


.    (276> 


'/ '  4 


and  comparing  this  with  the  general  polar  equation  of  a  com;  section 
referred  to  the  focus  as  a  pole,  viz. : 

.,  (i  -  *) 


r  = 


Is  +  e  cos  (a  +  <p)' 


we  find 


***i^n*ty=>  ■■■■  (-) 

And  this  last  value  will  be  greater  or  less  than  unity,  according  as  V* 

2  K       171 

is  greater  or  less  than 


r, 


Multiplying  and  dividing  the  last  factor  by  M,  r„  and  replacing  wi 
by  its  value,  the  orbit  will  be  an  ellipse,  parabola,  or  hyperbola,  ac- 
cording as 


212  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

K  v*  <  — --i ' '  M> ' r<  J 

r/ 

M, .  V;  >  **-= -  -Mt.rt. 

r? 

That  is,  according  as  the  living  force  of  the  primary  at  any  point  ot 
its  orbit  is  less  than,  equal  to,  or  greater  than  twice  the  work  its  rela- 
tive weight,  at  that  point,  would  perform  over  a  distance  equal  to  its 
radius  vector.  So  that  a  primary  may  describe  any  of  the  conic  sec- 
tions ;  as  well  as  the  ellipse,  the  only  condition  for  this  purpose  being 
an  adequate  value  for  its  velocity. 

Substituting  the  value  of  e%  in  Equation  (277),  we  find 

k.  .m.r.  .  .    , 

and  denoting  the  semi-parameter  by  p,  the  equation  of  the  curve  gives, 
by  making  a  -f-  <p  —  90°, 

4  c8         V,2 .  sin2  et .  r* 


and  denoting  the  semi-conjugate  axis  by  bn 


\  =  V«t-P  =  V<-*™e,-r,\/]^Z     •     •     •     (2?9)' 


b,  - 

k4 .  m 

Whence  it  appears  that  the  nature  of  the  orbit  and  its  transverse  axis 
are  independent  of  the  direction  of  the  primary's  motion,  while  the 
conjugate  axis  is  dependent  upon  this  element. 

§  201.— The  consequence  of  Kepler's  third  law  is  not  less  important. 
Denote  the  periodic  time  of  the  primary  by  Tf ;   then,  Equation  (264), 

7T .  at  bt 


T,= 


c       ' 


and    substituting   the    values    of  b„  m,  and  c,  Equations  (279)',  (273)', 

and  {275)', 

s 


T,  =  2n.a/t  .x/ I 


(M+Mt)k, 


MECHANICS    OF    SOLIDS.  213 

and  for  another  body  whose  mass  is  Mtn  about  the  same  central  body, 

.1 

and  by  division, 

t;  _«/   m+m„  *„      ' 

If  the  difference  of  the  masses  M,  and  Mu  be  so  small  in  comparison 
with  M  as  to  make  its  omission  insensible  to  ordinary  observation, 
which  is  the  case  in  the  solar  system,  the  above  may  be  written, 


t; 

rn    2   ~ 

t; 

rp    2 

_«/. 

"//• 


But  by  Kepler's  third  law, 


whence 

k,  ■=.  k/ 

That  is,  the  central  body  M  would  act  equally  on  the  unit  of  mass  of 
each  of  the  primaries  M,  and  M/A  were  they  at  the  same  distance ;  so 
that  not  only  is  the  law  of  the  central  force  the  same,  but  the  abso- 
lute force  at  the  same  distance  is  the  same,  and  it  is  one  and  the 
same  force  that  keeps  the  planets  in  their  orbits  about  the  sun. 

§202. — The  observations  of  Dr.  Maskelvne  on  the  fixed  stars,  show 
that  a  neighboring  mountain,  Schehallien,  drew  the  plumb-line  of  his 
instrument  sensibly  from  the  vertical :  and  those  of  Cavendish  and 
Baily  upon  leaden  and  other  balls,  demonstrate  this  power  of  attrac- 
tion to  reside  in  every  particle  of  matter  wherever  found  ;  and  that  it 
is  exerted  under  all  circumstances,  without  the  possibility  of  being  inter 
cepted.  It  is,  therefore,  concluded  that  matter  is  endowed  with  a  gen- 
eral gravitating  principle  by  which  every  particle  attracts  every  other 
particle,  and  according  to  the  law  before  given. 

PERTURBATI0N8. 

§  203. — Granting,   for    the    present,   that    universal    gravitation   is    a 
principle  of  nature,  and  denoting  the  distances  of  the  several   bodies  of 


214  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

the  system  from  the  central  by  r  with  subscript  accents  corresponding 
to  those  of  the  bodies  to  which  they  belong,  and  employing  the  same 
notation  in  regard  to  the  co-ordinates,  we  shall  have 

,*r*r\  /     M.M     x'  x' 

\4  4  i 

M     x"  M  x" 

it  '  il  4i 

M4Mlt x"-x' 

2  {M' M,)z  "     ?  (*"-x'H(y"-y?+t*" -Oa '  TW^WW^ffW^f 

which,  substituted  in  first  of  Equations  (258),  give 

but 

z"-x'  „         1        ,  1 

-  =  -r-7  •  d 


[(*"-  x'f+  (y "-  i/)2  +  (z'-zjy       dx'        V(x"-x?+(y"-y'?  +  (z'-z')* ' 
and  making 

x  =  s  — '     " /28n 

V(*''_a;')2  +  (y"-y')2+(*''-s')9     •    '    v      ' 

the  last  term  of  the  equation  above  becomes 

1     dX 

W/dx~" 
and 

—  -k\(M+M,).-i-l^Lr+K.-]  =  0. 
Make 

Jg=jf//(^,+yV,+^)  ,  if^WV«w  ,  ^     *  ;(2e2) 


r  3  r    3  if 


then  will 


rfi?      M„x"      Mm*>"  dX  ■  *"  dX 


rf*'~       r//3  >*„/  M,.dx'~         "   rj      Mt.dx" 

whieh,  substituted  above,  give,  after  treating  the  other  two  of  Equations 
^258)  in  the  same  way, 


MECHANICS    OF    SOLIDS.  215 

<Px'        .  r ,     x'        dR 


(283) 


The  curve  which  would  be  described  by  the  primary  about  the  central, 
under  the  reciprocal  action  of  thess  two  boJies  alone,  and  which  we 
have  seen  is  a  conic  section,  is  called  the  undisturbed  orbit  of  the  pri- 
mary. That  which  it  actually  describes  under  the  joint  action  of  all 
the  bodies  of  the  system,  is  called  the  disturbed  orbit.  The  undisturbed 
orbit  is  given  by  the  first  two  terms  of  Equations  (283) ;  the  disturbed 
by  all  three.  The  departures  of  the  disturbed  from  the  undisturbed 
orbit  are  called  2)erturbationsy  and  the  last  terms  of  Equations  (283), 
which  determine  them,  are  called  'perturbating  functions.  The  construc- 
tions of  the  perturbating  functions  are  given  in  Equations  (281)  and 
(282),  and  the  methods  of  computing  their  values  are  greatly  facilitated 
by  the  principle  of  the 

COEXISTENCE    AND    SUPERPOSITION    OF    SMALL    MOTIONS. 

§  204. — Denote  by  0„,  Qn4,  &c,  numerical  quantities  which  depend 
upon  the  perturbating  actions  of  the  bodies  whose  masses  are  Mit,  Mitl, 
&c,  and  of  which  the  values  are  so  small  as  to  justify  the  omission  oi 
all  terms  into  which  their  products  enter  as  factors,  in  comparison  with 
such  as  contain  them  singly.  The  co-ordinates  of  M4,  at  the  time  t, 
when  undisturbed,  being  x'  y'  z\  become,  when  the  body  Mt  is  disturbed 
by  Mlt  at  the  same  time, 

and  for  the  same  reason,  when  also  disturbed  by  Mltl, 

*  +  •>  +  9m  (*'  +  9,/V ;  jr*  +  e  J  +  bih  &  +  ej) ;  «'  +  9 J  +  9tt,{d  +  •,/) , 
or,  performing   the    multiplication    and    omitting   the    terms   containing 

*'  +  *'  (*„  +  BJ      y'  +  y'  (*„  +  0t„) ;    •'  +.  <  (*„  +  $„t) j 


y 


216 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


in  the  same  way,  when  also  disturbed  by  Mllit, 

^+*;(0„+0,„+0,L,);  y'+y'(0„+0„.+0„„);  *+<&,+*m+',,J 

and  for  the  simultaneous  disturbance  of  all  the  bodies  of  the  system, 

x'  +  x'zd^    y'  +  y'ZOy,    z'  +  z'zo,,- 

in   which   x'.Ed^,  y' .  2  0/y,  z',16^  are   the   increments  of  x' y' zf  re- 
spectively, due  to  the  joint  action  of  all  the  disturbing  bodies.     Now  let 

u  =<p(*'yV), 

in  which  <p  denotes  any  function  of  x' y'  z'.     Differentiating,  we  have 

du  du  d u 

and  performing  the  multiplications  indicated,  we  have 


d  u  =  i 


du       ,  .  d u        t-  du       ,  _ 


dz' 


du       t'  d  u       ,  _         r/  ?t       .  . 


+&"-+ 


;  +     &c. 


+ 


«kc.       + 


Whence  it  appears  that  the  perturbation  in  w  or  <p  {#'  y'  2'),  is  equal  to 
the  sum  of  the  separate  perturbations  due  to  each  of  the  perturbating 
bodies,  supposing  the  others  not  to  exist.  The  practical  effect  of  this 
principle  is  to  reduce  the  problem  of  the  perturbations  from  one  of 
s  vcral  to  one  of  a  single  perturbating  body,  and  to  give  rise  to  what 
is  known  as  the  problem  of  the  three  bodies,  viz. :  the  central,  primary, 
and  perturbating. 


UNIVERSAL     GRAVITATION. 


§  205. — From  all  of  which  it  is  manifest  that  either  Kepler's  laws 
cannot  be  rigorously  true,  or  universal  gravitation  is  not  a  Principle  ol 
Nature.      Now,  in  point  of  fact   observations  of  far  greater  nicety  than 


MECHANICS    OF    SOLIDS.  217 

those  01  Kepler  prove  that  his  laws  are  not  accurately  true,  though 
they  differ  but  slightly  from  the  truth ;  a  circumstance  arising  entirely 
from  the  fact  of  the  great  mass  of  the  sun  as  compared  with  the  sum 
of  the  masses  of  all  the  planets.  Were  there  but  a  single  body  in  ex- 
istence besides  the  sun,  it  would  describe  accurately  an  elliptical,  para- 
bolic, tv  hyperbolic  orbit  about  the  centre  of  the  sun,  depending  upon 
its  living  force  and  the  sun's  attraction.  A  third  body  would  derange 
this  motion  and  cause  a  departure  from  this  simple  path,  and  the  de- 
gree ^f  the  disturbance  would  depend  upon  the  mass,  distance,  and  di- 
rection of  the  disturbing  body  as  compared  with  those  of  the  sun.  The 
same  remark  would  apply  to  a  fourth,  fifth,  and  to  any  number  of  addi- 
tional bodies.  The  disturbed  orbits  in  the  solar  system  have  been  com- 
puted  by  Equations  (283),  and  the  complete  harmony  which,  is  found 
to  subsist  between  the  numerical  results  deduced  from  theory  and  ob- 
servation, is  the  strongest  possible  evidence  in  support  of  the  Law  of 
Uni'ersal  Gravitation. 

It  tLi;  principal  plane  of  the  solar  system,  as  determined  at  different 
and  remote  periods,  be  found  to  have  undergone  no  change,  this  will 
show  that  the  system  is  uninfluenced  by  the  action  of  the  fixed  stars 
and  other  distant  bodies,  and  its  centre  of  inertia  will,  §  183,  either  be 
at  rest  or  be  moving  uniformly  through  space  in  a  right  line ;  but  it 
the  principal  plane  be  found  to  have  changed  its  place,  it  will  be  a  sign 
that  the  system  is  in  motion,  and  that  its  centre  of  inertia  is  describing 
a  curvilinear  path  about  some  distant  centre. 

§  206. — Thus  much  for  the  larger  bodies  of  nature.  But  these  are 
themselves  built  up  of  innumerable  molecules  which  are  ever  on  the 
move  about  their  respective  places  of  relative  rest.  The  molecular 
forces  within  the  range  of  their  natural  action  vary  directly  as  the  dis- 
tance from  their  respective  centres.  Let  it  be  inquired  to  determine 
the  nature  of  the  orbits  under  thi&  law.    Then  will 

F=mk..r  = ; 

u 

which,  in  Equation  (272),  gives 

d*  u  k,.m 


218  ELEMENTS    OF    ANALYTICAL    MECHANICS, 

multiplying  by  2  d  u,  and  integrating,  we  find 

dA  +  «°  =  C-kf^ (284) 

da'  4  r  u  N 

from  which  we  get 

udu 


d  a  =  — 


/  _   .       k . .  m 


4  c 
the  negative  sign  being  taken,  because 


dU         W  dr    .  ;, (284y 


da         da  r*  da 

Placing  \  C9  —  \  C2  under  the  radical,  we  may  write 

1  —  2  udu 


da  —  %. 


v7?- 


Km,  (tf  - 1  cy 

4  c*    /     £!    ^•w> 

V  T        4ca 

and  integration, 

2  (a  +  <p)  =  cos  _1 


in  which  9  is  the  constant  of  integration. 

Taking  cosine  of  both  members,  replacing  u  by  its  value  and  solving 
with  respect  to  r,  we  find 


r  ac      .. 

Denote  by  r,  the  radius  vector  which  is  normal   to   the  orbit;   corre- 
sponding to  this  value  we  have 

du 
d  a 

and,  by  Equation  (284), 

1     Jt,.  m.r? 


MECHANICS    OB     SOLID8.  219 

and  because 

cos  2  (a  +  9)  =  cos2  (a  -f-  9)  —  sin*  (a  -f  <p),  ' 

the  above  reduces  to 


r  = 


I   1  mC        TfL      7* 

Vp cos>  +  *)  +  tJx?  '  sin'(a  +  9) 


.     .     (285) 


which  is  the  equation  of  an  ellipse  referred  to  its  centre  as  a  pole,  the 
semi-axes  being 


r,  and   —  \  • 


4 


rt        k,.  m 


§  2C7. — The  time  required  to  describe  the  entire  ellipse  being  deno- 
ted by  T,  we  have,  Equation  (264), 


tt  .  r, .  2  c  \  - 


m  k..m        n       a  /     1 

T  = '- —  =  2  7T  X ; 

r,.c  kt.m 

and  replacing  m  by  its  value,  Equation  (273)', 


T=2nX 


(M  +  M,)  kt 


(286) 


Thus  the  time  is  wholly  independent  of  the  dimensions  of  the  orbit, 
and  will  be  the  same  in  all  orbits,  great  and  small.  This  result  finds 
its  application  in  the  subject  of  acoustics,  thermotics,  optics,  &c. 

§  208. — Let  us  conclude  the  planetary  motions  with  the  centrifugal 
force  on  its  surface,  arising  from  the  rotation  of  one  of  these  bodies, 
say  the  earth,  about  its  axis. 

If  F,  denote  the  angular  velocity  of  a  body  about  a  centre,  then  wiU 
V=pVl,  and  Equation  (270)  becomes 

The  earth  revolves  about  its  axis  A  A'  once  in  twenty-four  hours, 
and    the    circumferences    of    the    parallels    of    latitude    have    velocities 


220 


ELEMENTS    OF    ANALYTICAL    MECHANICS, 


which  diminish  from  the  eqiator  to 
the  poles.     The  law  of  this  diminu- 
tion,   on    the    supposition    that    the     & 
planet  is  a  sphere,  is  given  by 

in  which  M  is  the  body's  mass,  Vy 
the  earth's  angular  velocity,  and  R' 
the  radius  of  one  of  its  parallels  of 
latitude. 

Denoting    the    equatorial    radius   C  E  =  C  P,  by  R,  and    the    angle 
C  P  C  =  P  C  E,  which  is  the  latitude  of  the  place,  by  <p,  we  have 

R'  =  R  cos  9  ; 

which  substituted  for  Rr  above,  gives 

F^MVfR  cosy (286)' 


The  only  variable  quantity  in  this  expression,  when  the  same  mass 
is  taken  from  one  latitude  to  another,  is  <p ;  whence  we  conclude  that 
the  centrifugal  force  varies  as  the  cosine  of  the  latitude. 

The  centrifugal  force  is  exerted  in  the  direction  of  the  radius  R'  of 
the  parallel  of  latitude,  and  therefore  in  a  direction  oblique  to  the  ho 
rizon   T  T'.     The  normal  and  tangential  components  are,  respectively, 

F, .  cos  cp  =  M  F,8  R  cos2  <p, 
Ft .  sin  9  r=  M  F,2  R .  sin  <p  cos  <p  =  \  M  V?  R  sin  2  <p  ; 

whence  we  conclude,  that  the  diminution  of  the  weights  of  bodies 
arising  from  the  centrifugal  force  at  the  earttis  surface,  varies  as  the 
square  of  the  cosine  of  the  latitude ;  and  that  all  bodies  are,  in  con- 
sequence of  the  centrifugal  force,  urged  towards  the  equator  by  a  force 
which  varies  as  the  sine  of  twice  the  latitude. 

At  the  equator  the  diminution  of  the  force  of  gravity  is  a  max- 
imum, and  equal  to  the  entire  centrifugal  force;  at  the  poles  it  is  zero. 
The  earth  is  not  perfectly  spherical,  and  all  observations  agree  in  de- 
monstrating that  it  is  protuberant  at  the  equator  and  flattened  at  the 
poles,  the  difference  between  the  equatorial  and  polar  diameters  being 
about  twenty -six  English  miles.     If  we  suppose  the  earth  to  have  beer 


MECHANICS    OF    SOLIDS.  221 

at  one  time  in  a  state  of  fluidity,  or  even  approaching  to  it,  its  present 
figure  is  readily  accounted  for  by  the  foregoing  considerations. 

To  find  the  value  of  the  centrifugal  force  at  the  equator,  make,  in 
Equation  (280)',  M=  1  and  cos  <p  =  1,  which  is  equivalent  to  suppo- 
sing a  unit  of  mass  on  the  equator,  and  we  have 

in  which,  if  the  known  radius  of  the  equator  and  angular  velocity  be 

substituted,  we  shall  find 

/ 
F4=  V*.R  =  0,  1112. 

To  find  the  anomlar  velocity  with  which  the  earth  should  rotate,  to 
make  the  centrifugal  force  of  a  body  at  the  equator  equal  to  its 
weight,  make 

/ 
^  =  32,  1937  =  F/2i?; 

in  which  32,  1937  is  the  force  of  gravity  at  the  equator. 
Dividing  the  second  by  the  first,  we  find 

32,1937       VJ*       — 

— =  ttt  =  289,  nearly  J 

0,1112        F,9  '  :i 

whence, 

that  is  to  say,  if  the  earth  were  to  revolve  seventeen  times  as  fast  aa 
it  does,  bodies  would  possess  no  weight  at  the  equator. 

IMPACT    OF    BODIES. 

§  209. — When  a  body  in  motion  comes  into  collision  with  another, 
cither  at  rest  or  in  motion,  an  impact  is  said  to  arise. 

The  action  and  reaction  which  take  place  between  two  bodies,  when 
pressed  together,  are  exerted  along  the  same  right  line,  perpendicular  to 
the  surfaces  of  both,  at  their  common  point  of  contact.  This  arises  frou. 
the  symmetrical  disposition  of  the  molecular  springs  about  this  line. 

When  the  motions  of  the  centres  of  inertia  of  the  two  bodies  are 
parallel  to  this  normal  before  collision,  the  impact  is  said  to  be  direct. 

When  this   normal    passes    through    the    centres    of   inertia   of  botk 


222 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


bodies,    and    the    motions    of   these    centres   are    along    that   line,    the 

impact     is    said    to     be     direct    and 

central. 

When  the  motion  of  the  centre 
of  inertia  of  one  of  the  bodies  is 
along  the  common  normal,  and  the 
normal  does  not  pass  through  the 
centre  of  inertia  of  the  other,  the 
impact  is  said  to  be  direct  and 
eccentric. 

When  the  path  described  by  the 
centre  of  inertia  of  one  of  the  bodies, 
makes  an  angle  with  this  normal, 
the  impact  is  said  to  be  oblique. 

When  two  bodies  come  into  col- 
lision, each  will  experience  a  pres- 
sure from  the  reaction  of  the  other;  and  as  all  bodies  are  more  or 
less  compressible,  this  pressure  will  produce  a  change  in  the  figure 
of  both  ;  the  change  of  figure  will  increase  till  the  instant  the  bodies 
cease  to  approach  each  other,  when  it  will  have  attained  its  maximum. 
The  molecular  springs  of  each  will  now  act  to  restore  the.  former 
figures,  the  bodies  will  repel  each  other,  and  finally  separate. 
•'  ".;i  .  Three  periods  must,  therefore,  be  distinguished,  viz.  :  1st.,  that 
occupied  by  the  process  of  compression  ;  2d.,  that  during  which  the 
greatest  compression  exists ;  3d.,  that  occupied  by  the  process,  as 
far  as  it  extends,  of  restoring  the  figures.  The  force  of  restitution 
must  also  be  distinguished  from  the  force  of  distortion ;  the  latter 
1  '-'''denotuig    the   reciprocal   action    exerted    between    the    bodies    in    the 

first,  and   the  former  in   the  third   period. 
"'  ••     The  greater  or  less  capacity  of  the    molecular  springs  of  a   body 

* 

'to    restore    to   it   the   figure    of    which    it    has   been    deprived  by   the 

:  '  "application   of  some    extraneous  force  when    the   latter  ceases  to  act, 

is  called   its  elasticity. 

"'  "  '   The  ratio   of  the   force  of    restitution    to    that    of  distortion,  is  the 

1  measure  of  a   body's  elasticity.     This    ratio    is    sometimes    called    the 

^-efficient  of  elasticity.       When    these    two    forces    are    equal,   the    ratio 


MECHANICS     OF     SOLIDS.  223 

is  unity,  and  the  body  is  said  to  be  perfectly  elastic;  when  the 
ratio  is  zero,  the  body  is  said  to  be  non  elastic.  .There  are  no  bodies 
that  satisfy  these  extreme  conditions,  all  being  raore  or  less  elastic, 
but   none  perfectly  so. 

Let  the  two  bodies  A  B  and  A'  B\  the  former  moving  along  the 
line  H  2\  and  the  latter  along 
IF  T\  come  into  collision  at  the 
point  0.  Through  0,  draw 
the  common  normal  N L.  De- 
note the  angle  H  G  N  by  9, 
and  H'  EN  by  9' — these  being 
the  angles  which  the  directions 
of  the  two  motions  make  with 
the    normal.      Also   denote  the 

velocity  and  mass  of  the  body  a 

AB  by  V  and  M  respectively,  and  the  velocity  and  mass  of  A'  B' 
by    V  and  M'. 

The  components  of  the  quantity  of  motion  of  the  two  bodies  in 
the  direction  of  the  normal  and  of  the  perpendicular  to  the  normal, 
will    be 

M  V  cos  9,     M'  V  cos  9'     and     M  V  sin  9,     M'  V  sin  <p\ 

The  former  of  these  components  will  alone  be  involved  in  the 
impact;  for  if  the  bodies  were  only  animated  by  the  latter,  they 
would  not  collide,  but  would  simply  move  the  one  by  the  other. 
For  simplicity,  let  the  body  A  B  be  spherical;  the  normal  will 
pass    through    its    centre  of  inertia. 

Denote  by  «,  the  velocity  of  the  body  A  B  in  the  direction  of 
the  normal  at  the  instant  of  greatest  compression,  and  by  u'  the 
velocity  of  the  body  A'  B'  at  the  same  instant  in  the  same  direction. 
Then    will 

Fcos  9  —  «,     and      V1  cos  9'  —  ur     •      •     •     (287) 
be   the  velocities  lost  and  gained  in  the  direction  of  the  normal,  and 
M(Veosq>  -  w),     and     W  (  V  cos  9'  -  u')    •  •  •  (288) 


224         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

be  the  forces  lost  and  gained  at  the  instant  of  greatest  compression ; 
and  hence, 

M  (  V  cos  <p  —  u)  -f  M'  ( V  cos  9'  —  u')  =  0 ;  •     •     (280) 

and  denoting  the  angular  velocity  of  the  body  A'  B'  by  P',  the 
distance  Of  D  from  the  centre  of  inertia  of  A'  B'  to  the  normal 
by  e,  and  the  principal  radius  of  gyration  of  A'  B',  with  reference 
to   the   instantaneous  axis  by  kt ,  then  will 

M(Vcoscp  -u).e 
V'    ~  M'k*      ~~  .  (290) 

and  since  the  velocity  u  must  be  equal  to  that  of  the  point  D  at 
she   end   of  the   lever   arm  e,  we   have 

u  =  u'  +  e.V/ (291) 

Substituting  the  values  of  u  and  u'  from  this  equation  successively 
In   Equation  (289),  we  find 

M  V  cos  <p  +  M'  V  cos  ©'  +  AT  e  V/ 
u=  — ttt-^^- '-*   •     •     (292) 


u    = 


M  +  M' 

M  V  cos  <p  -f-  M'  V  cos  ^  —  Me  Vf 
M  +  JT 


•         • 


(293) 


After  the  instant  of  greatest  compression,  the  molecular  springs 
of  the  bodies  will  be  exerted  to  restore  the  original  figures,  and 
tf  c  denote  the  co-efficient  of  elasticity,  then  will  the  velocities  lost 
by  A  B  and  gained  by  A'  B'  during  the  process  of  restitution  be, 
respectively, 

c  (  V  cos  9  —  w)     and     c  (P7  cos  <p'  —  u') ; 

and   the   entire   loss   of  AB,  and  gain  of  A'  B\  will  be,  respectively, 

Pcos 9  —  m  -f  c  ( V cos  9  —  w),    and     V  cos  9'  — w'  -f-  c  ( P' cos  9'—  »//). 

Also    the    gain    of    angular  velocity    of  the    body  A'  B',  during    th^ 
process  of  restitution,  will  be 

_  ,  (  Pcos  9  —  v) .  e  M 

(Vi  =c _ . 


MECHANICS    OF    SOLIDS.  225 

aud  the  whole  angular  velocity  produced  by  the  impact  and  denoted 
by    V0  will    be   given    by    the    equation, 

rf-{l+.4» ^— -L-.-g.      ■    .     ■      (294) 

Denoting  the  velocities  of  A  B  and  A'  B',  after  the  collision  b\ 
v  and  v',  and  the  angles  which  the  directions  of  these  velocities 
make  with  the  normal  by  &  and  $',  respectively,  then  will 

v  cos  d  =  Fcosp  —  V  cos  <p  +  u—c  (V  coscp  —  u)  —  ([  -f  c)  u  —  c  Pcos<p, 

t/cosd'  =  F'cosp—  F'cos<p'-|-w'— c(F'cos<p'  — u')  =  (\+c)u'— cF'cos<p',. 

and  replacing  the  values  of  w  and  w',  as  given  by  Equations  (29:2) 
and  (293), 

x  M  V cos q>  + AT  V  cos  <p' -}- AT  e  V'         Tr  t      \ 

vcosd  =  (l+c)  — - „   ,     __/ L-cVcos<p,   (295) 

t>'cosd'  =  (l+c)  U W^ ~  C  P  cos<p'  (290) 

Moreover,  because  the  effects  of  the  impact  arising  from  the  compo- 
nents of  the  quantities  of  motion  in  the  direction  of  the  normal  will 
be  wholly  in  that  direction,  the  components  of  the  quantities  of 
motion  before  and  after  the  impact  at  right  angles  to  the  normal  will 
be  the  same,  and  hence 

vsind  =  Fsin<p, (297) 

v'sind'  s-Fsbi'V (298) 

Squaring  Equations  (295)  and  (297)  and  adding;  also  Equations 
(296)  and  (298)  and  adding,  we  find  after  taking  square  root,  and 
reducing  by  the  relations 

cos2  6  +  sin2  4 .3=  Vi    cos2  6'  4-  sin2  6'  =  1  ; 


v=^[{l+cylZ^+^^'+M'etr''-cVcos*Y+  r**ym 


/r/,       xJfPcos<p4-3f'Fcos<p'— MeV/  n       imj-a^  /*am 

»»f=W[(l-|-e) r  '—  cF'cos<p']24-  *  2sin2<p -(300) 


*226    ELEMENTS  OF  ANALYTICAL  MECHANICS. 

Dividing  Equation  (297)   by  Equation  (295).  and    Equation  (298)  by 
Equation  (296),  we  have, 

;  V .  sin  9 

=  7"       .  M  Fcos©  +  M'  F'cos  a?  +  We  V!       ~~     ~'(301) 
(lo-r) ^qfT^ '--cVcos? 

f  __  P'.sin©' 

^^   ~~  77  ;     xJf  Fcos©  +  Mr  WooS?  -  Me  V'         "         '^302) 
(1+c) ' M+  M' cFcos©' 

Equations  (290)  and  (292),  will  give  the  values  of  u  and  V/,  in 
known  terms,  and  these  in  Equations  (294),  (295)  and  (296)  will 
give  the  values  of  V 0  v,  and  v\  and  all  the  circumstances  of  the 
collision  will  be  known. 

§210. — If  the  bodies  be  both  spherical,  then  will  e  =  0,  and  Equa- 
tion (294)  gives  Vt  =  0 ;  and  Equations  (299)  and  (300),  (301)  and 
(302).  become 

,=^[(l+e>*i^+*J  «"*-cV^tf+  F»ri.»9  •  •  •  (303, 

^^[(l+^^^/^-^-c^cos^+^si,^':.  (304) 

V  sin  <p  .  .  . 

tan^  =  trW i     nftTTf 7 *  •  (305) 

.  M Fcos  <p  -4-  M'  V  cos  or         _  v       ' 

(1  +  c) J-j, i-  -.  F  cos  , 

F'  sin  <p' 

ta„  »!  =  . Foo.,  +  irF^? ~ ;  •  •  (30'5) 

(1+e) Jf  +  if ^--c  F'cos,.' 

The  Equations  (303)  and  (304)  will  make  known  the  velocities, 
and  (305)  and  (306)  the  directions  in  which  the  bodies  will  move, 
after  the   impact. 

Now,  suppose  the  body  A'  B'  at  rest,  and  its  mass  so  great  that 

the  mass   of    A  B    is   insignificant    in    comparison,    then    will    V   be 

M 
eero,  M'  mav  be  written  for  M  4-  M'  and   -T7r  will  be  a  fraction  so 
'  *  M 


MECHANICS     OF     SOLIDS 


227 


small  that   all    the   terms    into    which  it   enters  as   a    factor    may    be 
neglected,  and  Equation  (303)    becomes 


v  =  V  -y/c2  cos2  (p  -|-  sin2  <p  ; 


and    Equation   (305), 


tan  6  =  — 


tan  ap 


(30?) 


The  tangent  of  &  being  negative,  shows  that  the  angle  JV  HK, 
which  the  direction  of  A  i?'s  motion 
makes  with  the  normal  N N'  after  the 
impact,  is  greater  than  90  degrees;  in 
other  words,  that  the  body  A  B  is 
driven  back  or  reflected  from  A'  B' . 
This  explains  why  it  is  that  a  cannon- 
ball,  stone,  or  other  body  thrown  ob- 
liquely against  the  surface  of  the  earth, 
will  rebound  several  times  before  it 
comes    to    rest. 

If  the  bodies  be  non-elastic,  or,  which  is  the  same  thing,  if  c  be 
zero,  the  tangent  of  &  becomes  infinite ;  that  is  to  say,  the  body 
A  B  will  move  along  the  tangent  plane,  or  if  the  body  A'  B'  were 
reduced  at  the  place  of  impact  to  a  smooth  plane,  the  body  A  h 
would   move   along   this   plane. 

If  the  body  were  perfectly  elastic,  or  if  c  were  equal  to  unity, 
which    expresses  this   condition,  then    would    Equation  (307)  become 


tan  0 


tan  (p 


(308) 


which  means  that  the  angle  NFTF=  E  H  N'  becomes  equal  to 
KHN'.  The  angle  E  H  N'  is  called  the  angle  of  incidence,  the 
angle  KHN',  commonly,  the  angle  of  reflection.  Whence  we  see, 
that  when  a  perfectly  elastic  body  is  thrown  against  a  smooth,  hard, 
and  fixed  plane,  the  angle  of  incidence  will  be  equal  to  the  angle 
of  reflection  . ,. 

Tf  the   angles  <p  and  <p'  be    zero,  then  will    cos  9  =  1,    cos  <p'  —   1, 


228  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

sin  <p  =  0,    sin  p'  =  0 ;    the    impact   will    be   direct   and   central,  and 
Equations  (303)  and  (304)  become 

,-g  1  C)MV+M'V'    ty 

,       ■    .,    .  MV+M-  V         v, 

+ -  0 1 ,&— ¥+*r  - tr  '> 

and  passing  to  the  limits,  non-elasticity  on  the  one   hand  and  perfect 
elasticity  On  the  other,  we  have  in  the  first  case,  c  =  0,  and 

M  V  +  M  V  /n^ 

V  =  -WTM- W 

M  V  +  AT  V 

■  '  =  -lrTJr- (310) 

and  in   the   second,  c  =  1,  consequently, 

ft  M  V  +  iT  F' 
v="-2-IrX-177--^ (311) 


M  +  M' 
MV  +  M>V> 

v  ~  z  ~WTm>        v 


•   •   • 


(312) 


CONSTRAINED   MOTION. 

§211. — Thus  far  we  have  only  discussed  the  subject  of  free  motion. 
We  now  come  to  constrained  motion. 

Motion  is  said  to  be  constrained  when  by  the  interposition  of 
some  rigid  surface  or  curve,  or  by  connection  with  some  one  or 
more  fixed  points,  a  body  is  compelled  to  pursue  a  path  different 
from    that   indicated   by   the   forces   which  impart   motion. 

§  212. — The  centre  of  inertia  of  a  body  may  be  made  to  con- 
tinue on  a  given  surface,  by  causing  it  to  slide  or  roll  upon  some 
other  rigid  surface. 

§  213. — We  have  seen,  §  128,  that  the  motion  ol  translation  of 
♦.ho   centre    of  inertia,    and    of  rotation    about    that    ooint,  are    whollv 


MKCLIANTCS     OF     SOLTDS. 


229 


independent  of  one  another,  arid  the  generality  of  any  discussion 
relating  to  the  former  will  not,  therefore,  be  affected  by  making, 
in    Equation  (40), 

o>  =  0;     8+  =  0;     8&  as  0; 
which  will  reduce  that  equation  to 

d2  x 
(2  P  cos  a  —  -— -  •  2  m)  0  xt 


+  (2Pos/3-  j^>Xm)8yt 


V==o. 


rf2^ 


4-  (2  P  cos  /  —  j-j  ,  Xm)  £  *j 

Making 

2  m  sa  M ;     2  P  cos  a  =  X;     2  Pcos  0  as  F;     2  /» cos/  =  Z; 
and  omitting  the  subscript  accents,  we  may  write 

Now,  assuming  the  movable  origin  at  the  centre  of  inertia,  and 
supposing  this  latter  point  constrained  to  move  on  the  surface  of 
which  the  equation   is 

L  =  F(xyz)  =  0, (314) 

the  virtual  velocity  must  lie  in  this  surface,  and  the  generality  of 
Equation  (313),  is  restricted  to  the  conditions  imposed  by  this  cir 
cumstance. 

Supposing  the  variables  x  y  z,  in  the  above  equations,  to  receive 
the  increments  or  decrements  8  x,  8  y,  8  z,  respectively,  we  have,  from 
the  principles  of  the  calculus, 


dL  dL  dL 

— —  o  x  -\ =—  •  o  y  -\-  -=—  •02  =  0. 

dx  dy  dz 


(315) 


Multiplying  \y  an  indeterminate  intensity  X,  and  adding   the   product 
to  Equation  (313),  there  will  result 

d2x  dL 


d?y        •       dL\  „ 


\  =  o. 


+ 


(*- 


M  • 


d2z  dL 


dtl 


+  X 


—  ) 

dzJ 


dz 


230 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


The  quantity  X,  being  entirely  arbitrary,  let  its  value  be  such  as  to 
reduce  the  coefficient  of  one  of  the  variables  8x.  d  y,  6  z,  say  tliat  of 
$x,  to  zero;  and  there  will  result 


X 


d2  x  d  L 


dt2 


d  x 


o, 


(316) 


and 


("  7>4v) »' +  <£*-g  +  >■£)  *  =  °-  <317> 


Now  in  Equation  (315),  5y  and  #2  may  be  assumed  arbitrarily,  and 
8x  will  result;  hence  8y  and  8 z  in  Equation  (317)  may  be  regarded 
as  independent  of  each  other,  and  by  the  principle  of  indeterminate 
coefficients, 


Y  - 

M. 

d2y 
dt2 

+ 

X- 

dL 
dy 

Z  - 

M- 

d2z 

+ 

X- 

dL 

o, 
o, 


(318) 


and  eliminating  X  by  means  of  Equation  (316),  we  find, 


M- 


-  M- 


d2y 
dt2 

d2z 


>• 


d2z\     dL 
T&)     ~dy 


dL 
d  x 

dL 

y 


(J 


-  M- 


-  M- 


d2x\     dL 
~d~t?)  '-d 

d2y\     d 
~~dt2)  '  ~d 


y 

IL 


=  0, 


=  0 


►  --(319) 


which,  with   the  equation   of  the  surface,  will   determine  the  place  of 
the   centre  of  inertia  at  the  end  of  a  given  time. 


MOTION   ON   A   CURVE   OF   DOUBLE   CURVATURE, 


§214. — If  the  centre  of  inertia  be  constrained  to  move  upon 
two  surfaces  at  the  same  time,  or,  which  is  the  same  thing,  upon 
a  curve  of  double   curvature  resulting  from  their  intersection,  take 


L  =  F(xyz)  =-.  0,   > 
H=F'(xyz)  =0tl 


(320) 


MECHANICS    OF     SOLIDS. 


231 


from  which,  by  the  process  of  differentiating  and  replacing  dx,  dy.   d z% 
by  the    projections  of  the    virtual    velocity, 


d  L  ~  4  L     t  d  L     „ 


a  a: 


rfy 


dg 


,         dH    r  tf#    , 

•  <J.r  +  --—  •  dy  +  — -  .d2  =  0. 
dx  dy  d  z 


(321) 


(322) 


Multiplying  the  first  of  these  by  X,  and  the  second  by  X',  adding  the 
products  to  Equation  (313),  and  collecting  the  coefficients  of  8xy  8 yt 
and  8  2,  we  have 


(_         %r  d2  x 


( 


+  (  Y  -  M 


dt2 

d2  y 

dt2 


dL 

dx 

dL 


+  y 


d  11 


d 
di 


-  )  Sx 
x  / 


f  (Z  -3/.^-  +  X 


dL         ^fdH\.. 
dy  dy  / 

dH 


y 


dt2 


dL 

dz 


+  X' 


dz 


) 


0     •  (323) 


Now  the  coefficients  of  twro  of  the  three  variables  8  x,  8  y  and  8  z, 
say  those  of  8x  and  8y,  may  be  made  equal  to  zero  by  assigning 
proper  values  for  that  purpose  to  the  indeterminate  intensities  X  and 
X',  in  which  case,  since  8  z  is  not  equal  to  zero,  its  coefficient  must 
also  be  equal  to  zero;    whence 


X  -  M 
Y  -  M 
Z  -  M 


d2x 
HI2 

<Py 
dt2 

d2z 


+  x 
+  x 

+  X 


dL 
d  x 

d  L 

dy 

dL 

d  z 


dH 


+ 

x'. 

dx 

0, 

+ 

x'. 

dH 

dy 

0, 

+ 

x'. 

dH 

0. 

dz 


(324) 


and  eliminating  X  and  X',  there  will  result 

d2x\      /d  L     dH 


(*t*-S)v( 


+  <*-*7»-:G! 


dz      d  y 
i  L     dH 


/ ' „        ,r  rf2z\      (d  L 


dx       dz 
d  L     dH 


dx 


dz  / 

if)   [rH825) 

d_H\ 
"7iV/ 


d_L 
dy 

dL     dH 

d  z 


dL     d_H 

dx       dy 


232 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


which,  with    he   equations  of  the    surfaces,  is   sufficient   to   determine 
the  co-ordinates   of  the   centre   of  inertia  when  the   time   is   given. 


(326) 


§215. — If    the    given    surfaces    be    the    projecting   cylinders   of    a 
curve  of  double  curvature,  then    will    Equations    (320)    become 

L  =z  F(xz)  =  0 ;  ) 
B  =  P  (y  z)  =  0  J I 

And   because  L  is  now  independent  of  y,  and   H  is  independent  of  xf 
we  have 

d  L  dH 

d  y  d  x  ' 

which  reduce  Equations  (324)  to 

V  TUT      d2x       I      1  dL  A 

X  —  M  •  -r-r  -f-  X  •  -y—  =  0; 


dt2 


dx 


r-Jf.^+X'.^O; 


dP 


dy 


_        .,    d2z    t    ^     dL        ^,    dH 
Z  -  M>  -—  +  X  .  -— -  +  X'  •  — — 


=  0; 


►     •       •    • 


(327) 


&ud  Equation   (325)  to 

/_        .,    c/2^\     dL    dH 
KX  —  M  >  —  —  I  .  — — 

V  dtz'       dz      dy 

V  dt2/       dx     d 


dL_  dH 

dx      d  z 


dL    dH 


>=  0. 


•         •         •         • 


(328) 


This,    with   the  equations   cf    the   curve,   will   give   the   place   of  the 
centre   of  inertia   at   the   end   of  a   given   time. 

§  216. — If  the  curve  be  plane,  the  co-ordinate  plane  x  z.  may  be 
assumed  to  coincide  with  that  of  the  curve ;  in  which  case  the 
second  of  Equations  (327),  becomes  independent  of  y,  that  vaiia 
ble   reducing  to   zero,   and 

d2y  =  0,     and     -,—  =  0: 
9  dy 


MECHANICS    OF    SOLIDS. 


233 


hence  Equations    (327),  bcome 
X  ■ 


d2x  dL 


r=  0; 

d*z  dL  dH 

d  P  dz  dz 


•     •     » 


=  0; 


(329) 


and   because   the   factor 


r-^  =  o, 


Equation    (328)    becomes,    on   dividing  out   the   common   factor 


dH 

dy1 


t*r         „     d2x\     dL         /_         ,_    d2z\     dL         „        ,„rtrtV 

§  217. — By    transposing  the  terms  involving  X,  in  Equations  (316) 
and    (318)  and  squaring   we   have 


(    r,  „  d2  Z\2 


The  second  member  of  this  equation  is,  Equation  (50),  the  square  of 
the  intensity  of  the  resultant  of  the  extraneous  forces  and  the  forces 
of  inertia.     Denoting   this  resultant   by  JV,  we  may  write 


V<£)'+  0"+  <&>'-'• 


•      •      • 


(331) 


«nd  dividing  each  of  the  equations 

dL 


d*x 


rf#  V  d&J 

dz  \  dt2/' 


234  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

obtained    by    the   transposition   just    referred    to,    by    Equation    (331). 
we  find, 


d  x 


X  -  M- 


J?  x 

~d& 


I  SdL  y       /d  L  \2       /, 


dLy 


d  L 

dy 


N 


Y  -  M- 


d2  y 


I  ML  \»       (d  L  \2        (d  L  V 

V  (77)  +  W?)  +  (-37) 


c/2 


JV 


if. 


>>  (332) 


^2 
rf*2 


/7^"Z  V2    "  rdL\l      /d  LV 

V  t©  +  (37)  *  Kti) 


N 


The  second  members  are  the  cosines  of  the  angles  which  the 
resultant  of  all  the  forces  including  those  of  inertia,  makes  with  the 
axes;  the  first  members  are  the  cosines  of  the  angles  which  the 
normal  to  the  surface  at  the  body's  place  makes  with  the  same  axes. 
These  being  equal,  with  contrary  signs,  it  follows  not  only  that  the 
forces  whose  intensities  are 


^©f©*(^ss 


are  equal,  but  that  the}'  are  both  normal  to  the  surface,  and  act  m 
opposite  directions.  The  second  is  the  direct  action  upon  the  surface; 
the  first  is  the  reaction  of  the  surface. 

Equation  (331),  will,  therefore,  give  the  value  of  a  passive 
resistance  sufficient  to  neutralize  all  action  in  the  system  which  is 
inconsistent  with  the  arbitrary  condition  imposed  upon  the  body's 
path.  If  the  body  be  constrained  to  move  on  a  rigid  surface  01 
line,    this    resistance    will    arise    from    its   reaction.     • 

§  218,— If  Equations    (332)    be    multiplied    by 

N, 
and  the   angles  which   the  u«ii*n.ul   re&Utance  <>!'  the  surface  makes  with 


MECHANICS    OF    SOLIDS.  235 

the    ixes   :r,  y,  2,    respectively,    be   denoted    by    8X,   Qpi   and    0„    those 
equations  will  take   the   form 


X  -  M  •  %?-  +  N-  cos  6    =  0 ; 

rf2  ?/ 
F-  Jlf -yf  +  iV-cos*,     =0; 

rf2Z 

Z  -  M  •  —  +  iV  •  cos  df     =  0. 
at1 


(333) 


§219. — To  impose  the  condition,  therefore,  that  a  body  in  motion 
shall  remain  on  a  rigid  surface,  is  equivalent  tc  introducing  into 
the  system  an  additional  force,  which  shall  be  equal  and  directly 
opposed  to  the  pressure  jpon  the  surface.  The  motion  may  then 
be  regarded  as  perfectly  free,  and  treated  accordingly.  The  same 
might  be  shown  from  Equations  (324)  to  be  equally  true  of  a 
rigid  curve,  but  the  principle  is  too  obvious  to  require  further 
elucidation. 

Equations  (333),  may,  therefore,  be  regarded  as  equally  appli- 
cable to  a  rigid  curve  of  anv  curvature,  as  to  a  surface;  the  nor- 
mal  reaction  of  the  curve  being  denoted  by  iV7,  and  the  angles 
which    If  makes   with    the   axes   x,  y,  ar,    by    0„  Qy,  and    &,. 

§  220. — To  find  the  value  of  JV,  eliminate  d  t  from  Equations 
(333),  by   the    relation 

1  V 


dt    ~  ds' 

in    which    V  and   s  are  the  velocity  and   the  space ;    t  ten   oy  transpo- 
sition   these    equations   may    be    written 

d'z  x 
iT.cos*.  =  M-  V*.-—  -  X-9 

(X  6* 

jV-cosd     =lf.  72--tI -:   Y\ 

tf.cos*,    =  M.  V*.^4  -  Z. 

as2 

15 


236 


Elements  of   analytical  mechanics. 


Squaring,   adding   and    reducing   by   the  relations 

R2  =  X2  -f  Y2  +  Z2, 
cos2^  -f-  cos2dj,   -f  cos2   6,  =  I. 


and    we   find 


J/2- 


N*  =   J 


^[(rf2*)2  +  (^y)2  +  (<*2z)2]  +  i?2 


Resolving  i?  into  two  components,  one  parallel  and  the  other  per- 
pendicular to  the  path,  the  former  will  be  in  equilibrio  with  the 
inertia  it  develops  in  the  direction  of  the  curve ;  and  denoting 
by    (p    the  inclination    of  R   to    the    radius   of  curvature,    we    have 

Rsm  y  —  M-  —  =  M-  V2- 


or, 


dt2 


0  =  /?.  sin  cp  -M  .  V2 


ds2 


d2s 

I72' 


Squaring   and    subtracting'   from    the  equation   above,  there   will   result 
V* 


N2=< 
but 


M'< 


—  •  ((d2x)2  +  {d2y)2  +  (d2z)2  -  {</**)*)  +  fi»W  D 

(X  d2x       V  d2y       Z    dU        .        d2s\ 
\~R"d72+R'ds~2  ^  R'ds2~  S'n  9  '  if*2' 


X    dx       Y   dy       Z    dz 

sin  o  =  —  •        4-  —  •  — -  n •  —  • 

*       R   d*        R   ds  T  R   ds> 


multiplying    the    second     member    by    p  -.-  p,    substituting   above,    and 
reducing   by    the    relations, 


dx  dy 

d2x     dx  d2s        ds  d2y     dy  cPs        ds 

ds2      dsc/s2~     d*'  d.s2      ds  ds2~     ds 


dz 


a 


cos  <p  = 


X       ds       Y     1 7s      Z      "ds 


d2z     dz  d2s        ds 
ds2     d.s  ds2'      dt 

dz   * 


R'?    ds       R 


•See  Appendix  No  2. 


MECHANICS    OF    SOLIDS. 


237 


and 


ds2 


y/{d?  x)2  +  (d2  y)2  +  {d2  zf  -  {<*>*)* 


in    which    p    denotes   the    radius   of  curvature,    we   have. 


r*       m  v2 

N2  =  M2-  —  —  2 Rcostp  -f  i*2Cos2<p; 

P  P 


and  taking  square  root, 


N  = 


MV2 


R  cos  <p. 


(334) 


The  first   term   of  the   second    member  is, 

§  195,    the    centrifugal    force    arising   from 

the  deflecting  action  of  the  curve,  and  the 

last  term  is  the  normal  component  of  the 

resultant   R.     As  the   equation    stands,   its 

signs  apply  to  the  case  in  which  the  body 

is  on   the   concave   side  of  the   curve,  and 

the   resultant   acts   from  the   curve.     The  angle  <p,  must   be   measured. 

trom  the   radius   of  curvature,  or   that   radius   produced,  according  as 

the  body    is    on    the  concave   or   convex    side  of   the   curve.      When 

the    body   is    moving   on    the   convex    side   of    the    curve,    the    first 

term     of    the    second    member    must    change    its    sign    and    become 

negative. 

§221. — Writing  Equations  (333)  under  the  form 

dP-x 

1  M-j^=Z  +  JVcostf,; 

multiplying  the  first  by  2rfar,  the  second  by  2c?y,  the  third  by  2rfs, 
adding  ani  reducing  by  the  relation 

.     /dx  dy  .  dz  \  _ 

rfn-'  cos  8,  -+-   —  •  eosfl     -f — —  •  cos  I L  I  •  ss  0, 

>■</  s  an  '  a  ,v  / 


238  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

the  second  factor  being  the  cosine  of  the  angle  made  by  the  nor 
mai    and    tangent   to    the   curve,  we  have 

/2  dx  .  d2  x  4-  2  dy  .  d2y  4-  2dz  •  d2 z\ 
M-  ( : *d         y  ~  )  =2{Xdx+Tdy+Zdz); 

integrating  and  reducing  by 

dx2  -f  dy2  4-  dz2 
~  I?  ' 

we  find 

M  V2  z=  2  f(Xdx  +  Ydy  +  Zdz)  +  a     •    •     (335) 

This  being  independent  of  the  reaction  of  the  curve,  it  can  have  no 
effect   upon    the    velocity. 

If  the   incessant   forces    be   zero,  then  will 

X  =  0 ;     r  =  0;    and    Z  =  0 ; 
and 

0 

M> 

that  is,  a  body  moving  upon  a  rigid  surface  or  curve,  and  not  acted 
upon  by  incessant  forces,  will  preserve  its  velocity  constant,  and  the 
motion  will    be    uniform. 

We  also  recognize,  in  Equation  (335),  the  general  theorem  of 
the  living  force  and  quantity  of  work ;  and  from  which,  as  before, 
it  appears  that  the  velocity  is  wholly  independent  of  the  path  de- 
scribed. 

Example  I. — Let  the  body  be  required  to  move  upon  the  interior 
surface  of  a  spherical  bowl,  under  the  action  of  its  own  weight.  In 
this   case, 

L  =  x2 ■  +  y2  +  z2  -  a2  =  0 ;      .     .     .     .     (33t>) 

dL  dL  dL 

-j—  =  2  x ;     — -  =  2  y  ;     -r—  =  2  z : 

dx  dy  *  '      dz  ' 


MECHANICS     OF     SOLIDS 


239 


and  the  axis  of  z  being  vertical    and 
positive    downwards, 

which    values    in     Equations    (319). 
give 

d2 x  d?y         .    "| 


gy  -y 


(Pz 

dt2 


+  * 


d2y 
dt2 


^•(337) 


0; 


and  differentiating  the  equation  of  the 
sphere  twice,  we    have 

xcPx  4*  yd2y  +  z .  d2  z  =  —  (dx2  -f  d  y2  +  d  z2)  \ 

dividing    by  d t2,  and  replacing  the    second  member  by  its  value    F2, 
(he    velocity,  we  find, 


dt2 
But,  Equation  (335), 


(P-x  d2y  d2  z  T_, 


dt2 


dt2 


» 


V2  =  2gz  +  C 


(ms) 


and  denoting  by  F'  and  kt  the  initial  values  of  V  and  2,  respectively, 
we   have 


F2  =  F'2  +  2ff  (z  -  £), 
wh:ch   substituted   above,  gives 

^#  d?V  d?z         ~     /,  v        »t 

+  y~  +  *•— --=2<7(A:-2)    -   F 


df*2 


dt2 


dP 


(339) 


Eliminate  ar,  y,  c?2#,  c?2y,  from   this   equation    by    means  of  Equa- 
tions (336)  and  (337). 

From    the   latter  we  find,  ' 


<Py     '    y_  (<&z_         \ 
dfi  z    \dt2     '  9) 


d2x 


x  x    /dzz  \ 

~t?    =  T  \dT  ~  9) 


,:+ 


240  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

which  substituted  in  Equation  (339),  and  reducing  by  means  of 
Equation  (336),  we  get 

d2z 
a2-—  =  g(a?  -  3z2  +  2kz)  -  V*z\ 

multiplying  by  2dz,  and  integrating,  we  find 

a2-  ~  =  2ff  (a2z  -  z3  +  kz2)  ~  v'2  z2  H   r : 

in  which  C  is  the  constant  of  integration,  and  to  determine  which, 
we  denote  the  component  of  the  velocity  V\  in  the  direction  of  the 
axis  z,  by  Vt\   and   make   z  =  k.     This  being  done,  we  get 

c  =  a2. ra'2  +  f*jp  -  $#***; 

whence, 

a2  •  Sg-  =  2#  («2z  -  z3  +  #z2)  —  F'2z2  +  a2  Fr'2  +  V*k*  -  2^a2*, 

Adding  and  subtracting  a2  V'2  in  the  second  member,  this   reduces  to 

♦n  which 

(7,     =     (a2  -  k2)  F'2  -  a2  F/2. 

Finding   the   value  of  d  t,  and  integrating,  we  have 

/adz  , 

,  ....  (340) 

VV  -  *2)  [  F'2  _  2^  (*-*)]-  C, 

Could  this  equation  be  integrated  in  finite  terms,  then  would  z 
become  known  for  a  given  value  of  t\  and  this  value  of  z  in 
Equation  (336),  and  the  first  of  Equations  (337),  after  integration, 
would  make  known  the  values  of  x  and  y,  and  hence  the  position 
of~ihe  body ;  its  velocity  would  be  known  from  Equation  (335V 
But   this   integration    is   not   possible. 


MECHANICS    OF    SOLIDS.  241 

§222. — We  may,  however,  approximate  to  the  lesult  when  tho 
initial  impulse  is  small  and  in  a  horizontal  direction,  and  the  point 
of  departure  is  near  the  bottom  of  the  bowl.  Let  &  be  the  angle 
which  the  radius  drawn  to  the  variable  position  of  the  body  makes 
with  the  axis  of  z;  <p,  the  angle  which  the  plane  of  the  angle  6 
makes  with  the  plane  through  the  axis  z  and  initial  place  of  the 
body'  supposed  in  the  plane  xz  ;  V  =  (3  yfya,  the  velocity  of  pr«>; 
jection  in  a  horizontal  direction,  (3  being  a  very  small  quantity ; 
and   a   the   initial   value  of  6.     Then,  because 

z  =  a  .  cos  &  =  a  (1  —  2  sin2  \ff)\     k  =  a,  cos  a  =  a  (1  —  2  sin9  £  a)  ; 

d  z  —  —  a  .  sin  6  .  d  d  ;      Vt'  =  0  ; 

Gt  —  (i  ■  4  sin*  \  a  .  cos'  i  a  .  /3* .  a //  =  a  .  ji  .  <y .  sin'  «  ; 
Equation  (340)  becomes 

sin  &  .  db 


*   ~a  J  V8  (sin2 


#  ^  \/j3  (sin2  d  —  sin"  a)  —  4  sin*  A  (sin'  J  d  —  sin*  ^  a) 

- 

and  making  &   and  a   very   small,   their    arcs    may    be    taken    for   their 
sines,  and  the  above  becomes,  after  differentiating, 

^  JZ.-^-'  .    .     (341) 

d '  V    g        ^(a?  _  02)  (02   _  £2) 

which  may  be  put  under   the   form 

-46.dfi 
2t  = 


V  g  J  JTtf^ 


^/(a2   _   £2)2   _    [2d2   -    (a2   +   /D^)]3 

whence,  by   integration, 

making     *  =  0,    and     6  —  a.     we     have     C  =  —  cos    1  .  y^a  ~-  i/#*     (" 
(7=0;    and    solving    the  equation    with    reference   to   d,    we  get 

6*  =  i  (a*  +  /3')  +  J  (a*  -  /3*) .  ccs2  y/|  •  t.      •  •  (343) 


242  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

From  which  it  appears  that  the  greatest  and  least  values  of  d 
will  occur  periodically,  and  at  equal  intervals  of  time.  The  formel 
of  these  ■  values   is   found    by   making 


,cos 


2 kJ9- >t  =  1 ;     whence  2 \J- - 1  =  0,     or  =  2 *r,     orir4ir, 


arid  so  on;    and  for   a  single  interval  between  two  consecutive  maxi- 
ma,- without   respect   to   sign, 


A" 

t  —  f\/-;  •    •    • (344) 


the  maximum  being  a. 

The   least   value   occurs    when 

cos  2  v/-  •  t  =  —  1,     or  2 \  /- •  £  =  <r,  or  =  3  r.    &c. 

V  a  V  a 

whence  for  a  single    interval   between  any  maximum  and  the  succeed- 
ing minimum, 

...-,     .,;  '  =  i*V7; *     (345) 

the    minimum    being   /3. 

The  movement  by  which  these  recurring  values  are  brought  about, 
is  called  oscillatory  motion;  that  between  any  two  equal  values  is 
called  an  oscillation ;  and  when  the  oscillations  are  performed  in 
equal    times,    they   are    said    to    be   Isochronous. 

Again, 

dy        dq>    dtt 


d&         dt     d& 


suostituting  for  —=- ,  its  value  obtained  from  the  relation  y  —  ztanip, 


we  find 


rf<p  1  /       dy  dx\     dt 

dl      '  x2  4-  v2      ^    '  ~dt    ~~  V  *  Til  '  T&' 


Integrating  the  first  of  Equations  (337).  we  get 


_^_y.__  =  2c=  F'ax  =  a/3a  ^g  a  \ 


X    dt       *    dt 


MECHANICS    OF    SOLIDS. 


243 


substituting    this   above,  and  also  the   value   of  —r-r »  given  by  Equa- 
tion (341),  we  find 


;  > 


««  ^(a2  —  d2)(02  —  j8») 

dividing  this  by  Equation  (341), 

a.  /3 


•     •     •    • 


(346) 


d(p  _     /£   <*-$  fi_ 

dt      ~\    a'    d*        "V  a  ' 


i  (a2  +  /32)  +  £  (a2  -  0*)  .  cos2  i/-  •  ' 


but 


cos  2  \I—  •  t  =  cos2  '\/—  •  <  —  sin2  \I-  •  * ; 


whence 


g   =  ^£ —^ — -  ;   •   •   •  (34^ 

a2  •  cos2\  /—  •  /  4-  /32 .  sin2\  /—  •  t 


a2.cos2\/— ./  +  /32.sin2 

a  .       v    a 


from  which  we  find 


9  >dt 


0  V   a 

a '      .    fg 

cos2  \  /  —  •  t 

,  V    a 

d<p  =  — 


^  .„„, .  A 


? 


1  +  i— .tan2\/-^-.* 
a2  V    a 

integrating,    and  taking    tangents   of  both    members, 

tan  <p  =  —  •  tan  \J  —  •  t (348) 

from  which   the   azimuth    of  the   plane    of    oscillation   may   be   found 
at   the   end   of  any  time. 

Making   tan  <p  =  oo,  we   have 

7  1  3  5 

-.,  =  -.;     or     =  -,;     or     =T*,&af 


244  ELEMENTS'   O'F    ANALYTICAL    MECHANICS. 

and   the   interval  from  the  epoch  to   the   first  azimuth  of  90°,  is 

1  fa 

and   to   the   first   azimuth   of  270°, 

3  fa 

*«  =  -** -\r 

and  the  interval  from  the  azimuth  of  90°  to  the  next  azimith  of  270° 

<,,-',  =  <  =  *-\/y> 

equal   to   the   time  of  one   entire   oscillation. 

From    Equation    (348)    we    have,    after   substituting    for   tanp    its 
* alue   in   the   relation   y  =  x  tan  <p, 


ft 

adding    mity  to   both   members, 


SS-  =  tan2  \Pa- ' ' 


also    from   y  =.  x .  tan  <p, 

x2  +  y2 


x2 


=  1  -f  tan2  <p  ; 


dividing   the  last  equation  by  this  one,  and  replacing  x2  -\-  y2  by  its 
value  a2  —  z2,  from    the   equation   of  the   surface,  we  get 

1  +  tan2  \/—  •  t 

a2y2  +  (32X2  =  &'!'(&  -  z2).  2  V    a ; 

1  +  tan2  m 

but,  neglecting  the  term  involving  6\ 

a2  —  z2  =  a262; 

substituting   this    above,  replacing    tan2(p    by    its    value    in    Equation 
(348),  and  fl2  by  its  value   in  Equation  (343),  after  making 

cos  2  V  if  t  =  1  -  2  sin2  y9--  i 
a  a 


MECHANICS    OF    SOLIDS 


245 


and    reducing   by    the   relation, 


sin* 


a 


9 


we    have 


tan2  \  -t 
a 

I  +tan2|/£.* 


x*         y6 

^2     +    ]Q2 


=  a 


2. 


•        •       • 


(349) 


which  shows  that  the  projection  of  1he  path  of  the  body  on  the 
plane  xy,  is  an  ellipse  whose  centre  is  on  the  vertical  radius  of  the 
sphere,  and  that  the  line  connecting  the  body  with  the  centre  of 
the   sphere,  describes   a   conical   surface. 

If  a  =  /3,  then  will,  Equations  (343)  and  (348), 


&2  =  a2  =  J8»;     <p  =  \/—'t; 


9 


a 


and,  Equation    (349), 


x2  +  y2  =  a?  a2 ; (350) 

hence,    the    body    will    describe    a    horizontal    circle    with    a    uniform 
motion. 

The   pressure  upon    the    surface,  at  any  point  of  the    body's  path, 
is    given  by  the  value  of  N  in  Equation  (334). 

§223. — Example  2. — Let  the  body,  still  reduced  to  its  centre  of 
inertia  and  acted  upon  by  its 
own  weight,  be  also  repelled 
from  the  bottom  point  A 
of  the  bowl,  by  a  force  which 
varies  inversely  as  the  square 
of  the  distance  ;  required  the 
position  of  the  body  in  which 
it  would   remain  at   rest. 

As  the  body  is  to  be  at 
rest,  there  will  be  no  inertia 
exerted,  and  we  have 

d2x        _       cPy         _        (Pz 

—  o  •     ~  =  o  •      —  0  • 

dfi        V>      dfi  '      dfi   ~U> 


246  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and  assuming  the  axis  z  vertical,  positive  upwards,  and  the  origin 
at   the  lowest  point  A, 

L  =  x2  +  y*  +  #  -  2az  =  0,     .     .     .     .     (351) 
dL        «         dL        <>         dL 

and  denoting  the  distance  of  the  body  from  the  lowest  point  by  r. 
the  intensity  of  the  repelling  force  at  the  unit's  distance  by  F,  and 
the   force    at   any    distance  by  P,  then  will 

F 
P-^>     r  =   V*2  +  V2  +  *2;  •     •     •     •     (352) 

'7*  ii  *9 

for    the  force  P.    cos  a  =   — ;     cos  ft  =  — ;     cos  y  =  —  ;     for     the 

r  r  '  r 

weight  Mg,  cos  a'  =  0;     cos  ft'  =  0 ;     cos  y'  =  —  1  ;    and 

Fx  Fy  Fz 

These   several  values  being  substituted  in  Equations  (319),  give 

Fyx        Fyx 
r3  r3     ~  u> 

/Fz  \  Fv 

(73-  -M9)-V-1f.(z-a)  =  0. 

The  first  equation  establishes  no  relation  between  x  and  y,  since 
the  equilibrium,  which  depends  upon  the  distance  of  the  particle 
from  the  source  of  repulsion,  would  obviously  exist  at  any  point 
of  a  horizontal  circle  whose  circumferenco  is  at  the  proper  height 
from   the  bottom. 

From    the   second   equation   we  deduce, 

Fa  M 

-73-   =   ** 


(Fa\\ 
-   \~Mg)    ' 

F         r3 


•     •     •     •     • 


Mg        a  (353> 


MECHANICS     OF     SOLIDb.  247 

from  which  r  oecomes  known ;  and  to  determine  the  position  of  the 
circle  upon  which  the  body  must  be  placed,  we  have,  by  makin« 
x  =  0  in  Equations  (352)  and  (351), 

y^2  +  y3  =  r, 
f  _j_  Z2  _  2  a  z  _  0 

Equation  (353)  makes  known  the  relation  b*  tween  the  weight 
of  the  body  and  the  repulsive  force  at  the  unit's  distance;  the  in- 
tensity  of  the  force  at  any  other  distance  may  therefore  be  deter- 
mined. 

If  there  be  substituted  a  repulsive  force  of  different  intensity, 
but  whose  law  of  variation  is  the  same,  we  should  have,,  in  like 
manner, 

F  r'3 


Mg    ~    a   ' 
hei.ce, 

F :  F  :  :   r3  :  r'3 ; 

that  is,  the  forces  are  as  the  cubes  of  the  distances  at  which  the 
body   is   brought    to   rest. 

If,  instead  of  being  supported  on  the  surface  of  a  sphere,  the 
body  had  been  connected  by  a  perfectly  light  and  inflexible  line 
with  the  centre  of  the  sphere  and  the  surface  removed,  the  result 
would  have  been  the  same.  In  this  form  of  the  proposition,  we 
nave   the  common  Electroscope. 

The  differential  co-efficients  of  the  second  order,  or  the  terms  which 
measure  the  force  of  inertia,  being  equal  to  zero,  Equations  (332), 
show  that  the  resultant  of  the  extraneous  forces,  in  this  case  the 
weight  and  repulsion,  is  normal  to  the  surface,  which  should  be  the 
case ;  for  then  there  is  no  reason  why  the  body  should  move  in 
one  direction  rather  than  another.  The  pressure  upon  the  surface  is 
given   by   the   value   of  iV,  in  Equation   (334). 

g  224. — Example  3.     Let   it   be   required  to  find  the  circumstances 


248  ELEMENTS     OF     ANALYTICAL    MECHANICS. 


of  motion   of  a  body    acted    upon   by    its   own   weight  while   on   the 
arc  of  a  cycloid,  of  which 
the   plane   is    vertical,  and 
directrix  horizontal. 

Taking  the  axis  of  2, 
vertical;  the  plane  zx,  in 
the  plane  of  the  curve; 
and  the  origin  at  the  low- 
est  point,  then    will 


/ -1  z 

L  =  x  —\/2az  —  z2  —  a  versin     — =0;  •     •    (354) 


in   which   z   is   taken   positive    upwards. 


dL        ,       dL 

=  i; 


d  x 


dz 


-V- 


(355) 


X  =  0  ;     Z  =  —  Mg, 


and  Equation  (330)  becomes 


d2x         fa  a  —z  d2z 

tW-T-   +*  +  — 


dt2 


(350) 


and    by    transposition    and    division, 


d2x 
dt2 


9 


d2z 


12  a  —  z       dt2       ,2a  —  z 


(357) 


From    the    equation    of  the   curve    we    find, 


2dx  =  2dz- 


it. 


2a  —  z 


•  •     • 


(353) 


multiplying    by   Equation    (357),  there    wi  1    result 


2d  x  .  d2  x 

Ti2 


==  -  2gdz  — 


2dz.d2z 
dt2 


MECHANICS    OF    SOLIDS.  219 

and    by    integration, 


<*• 


d  x2  -f  dz* 


and   supposing    the    velocity   zero,    when   z  =.  k-y 

0  =  C—*Zgh; 
which   subtracted    from    the    above   gives 

dx2  -\-  dz2 


=  2g(k-s)',.     .     .  .359) 


dt2 
and    eliminating   dx2    by    means    of  Equation    (358), 

dz2        9    /I  2\ 

— —  =  -  •  ih  z  —  zl) 

dt2        a    y  ' 

whence, 

dz 


dt 


-Vi- 


y/h  Z  —  2a 


the    negative    sign    being    taken    because   z   is   a  decreasing    function 
of   t. 

By   integration, 

/a      f        dz  fa~  .  -i  2* 

t  ~  —  1/  —  *  /  — .  =  —  \/ versm     *-—  +  V. 

V   9    J    y/hz  -  z2  *   9  * 

Making  z  =  A,  we  have 

0  =  —  \J  —  *  versin~  '  %  -+•  C ', 


250  ELEMENTS    OF    ANALYTICAL    MECHANICS, 

whence, 


and 


«M 


.  -1  2zN 
versin 


•  -~) (3(5°) 

When   the    body  has  reached  the  bottom,  then  will  e  ~  0,  and 

t  =  *  \/— i 

which  is  wholly  independent  of  k,  or  the  point  of  departure,  and 
we  hence  infer  that  the  time  of  descent  to  the  lowest  point  will  r>e 
the  same  in  the  same  cycloid,  no  matter  from  what  point  the  body 
starts. 

Whenever  t  =  &,  the  body  will,  Equation  (350),  stop,  and  we 
shall    have    the  times   arranged  in  order   before   and    after  the   epoch, 

~4*V75   ~2*V7;  0;   2Vt*;   4*V7'  &c" 

the    difference   between  any  two   consecutive  values   being 

2  nt  v/— • 
V  y 

The  body  will,  therefore,  oscillate  back  and  forth,  in  equal  times. 
The   cycloid  is   a  Tautochrone* 

The    pressure   upon    the   curve   is   given    by  Equation  (334). 

The  time  being  given  and  substituted  in  Equation  (360),  the  value 
of  z  becomes  known,  and  this,  in  Equations  (359)  and  (354),  will 
give    the   body's  velocity  and.  place. 

§225. — Example  4. — Let  a  body  reduced  to  its  centre  of  inertia, 
and  whose  weight  is  denoted  by  W*  be  supported  by  the  action 
of  a  constant  foree  upon  the  branch  EH  of  an  hyperbola,  of  which 
the  transverse  axis  is  vertical,  the  force  being  directed  to  the  centre 
of  the    curve.      Required    the    position   of  equilibrium. 


MECHANICS    OF     SOLIDS, 


251 


Denote  the  constant  fcrce  by  W\  which  may  be  a  weight  at  the 
end  of  a  cord  passing  over  a  small  wheel 
at  C,  and  attached  to  the  body  M.  De- 
note the  distance  CM  by  r,  and  the  axes 
of  the  curve  by  A  and  B.  Take  the  axis 
z  vertical,  and  the  curve  in  the  plane  xz. 
Make 

P '  =  W, 
P"  =  W 


tnen  will 


cosy'  =  1,     cos  a'  =  0, 


H  %  It  <    * 

cos  y"  = »     cos  a     = , 

'  r  r 


X  =  F  cos  a'  +  P"  cos  a"  = 


r 


Z  =  Pf  cos  7'  +  P"  cos  7"  ss  W  -  W .  —  i 
and    as   the    question    relates  to  the   state   of  rest, 

The   Equation   of  tho   curve   is 

'  L  =  A2x*  -  B2z2  +  A2B2  =  0 ; 

whence, 

dL 


dx 

dL 
dz 


sb  2^2ar, 


=  -2B2z; 


these  values  substituted   in  Equation  (330),  give 


whence, 


W'B*  —  -  WA2x  +   W'A2  —  ss  0- 
r  r 


(42  -f  £2)  JT  •*  -  f*M2r  =  0 
16 


(361) 


252 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


But 


i«2   — -    -r2 


X2  -f  z2  =  z2  + 


B2 

A2 


A2  4-  B2 


whence,  denoting  the  eccentricity  by  c, 

r  =  ^/e2z2  -  B2 
and   this,    in  Equation   (361),  gives  after  reduction, 

B  .  W 


z  = 


e(W2-  W'2  erf 


which,  with  the  equation  of  the  curve,  will  give  the  positirii  o^ 
equilibrium. 

If  We  be  greater  than  W,  the  equilibrium  will  be  mi^ja&ibie- 
If  We  =   W,  the   body  will  be  supported  upon  the  asymptote. 

The  pressure  upon  the  curve  is  given  by  Equation  (334). 

§  226. — Example  5. — Required  the  circumstances  of  motion  of  a 
body  moving  from  rest  under  the  action  of  its  own  weight  upon  an 
inclined   right   line. 

Take  the  axis  of  z  vertical, 
the  plane  z  x  to  contain  the 
line,  and  the  origin  at  the 
point  of  departure,  and  let  z 
be  reckoned  positive  down- 
wards.    Then  will 


L  =  z  —  a  x  =  0, 


d  L 
dz 


.       dL 
=  1;    -y-  =  —  a; 
d  x 


which   in  Equation    (330)  give,  after  omitting  the    sommon  factor  Ms 

d2x  d2z 

--dfi+a^~aT^  =  Q- (362) 

From  the  equation  of  the  line  we  have 


d*x  m 


(Pz 


MECHANICS    OF    SOLIDS.  253 


which  in  Equation  (362),  after  slight  reduction, 

d2z  _       a2 

dt2    ~   Ifo2 

Multiplying  by  2dz,  and   integrating, 


'9' 


dz2  a2 

~dJ  =      9  \  +  a 

the   constant    of  integration    being   zero. 
Whence 


2     " 


/2(1  +  a2)       dz 
V        #  •  a2  2  -/  z 

and 

'        ,  =  ^3E37,=^+^;    .    .    .   (3C3) 
V        ga2  y       g  azz 

the  constant   of  integration  being  again  zero. 

The   body  being    supposed    at  B,  then  will    z  =  AD-,    and    if  we 
draw  from  B  the  perpendicular  B  C  to  ^4  B,  we  have  . 


2  .  2 


^.g  1  4-  a 


z2  .      «2      ' 


which  substituted  above, 


(364) 


in    which  of  denotes  the  distance  A  C. 

But  the  second  member  is  the  time  of  falling  freely  through  the 
vertical  distance  d\  if,  therefore,  a  circle  be  described  upon  A  C  a-> 
a  diameter,  we  see  that  the  time  down  any  one  of  its  chords,  ter- 
minating at  the  upper  or  lower  point  of  this  diameter,  will  be  the 
same  as  that  through  the  vertical  diameter  itself.  This  is  called  the 
mechanical    property  of  the  circle. 

Example  6. — A  spherical  body  placed  on  a  plane  inclined  to  the 
horizon,  would,  in  the  absence  of  friction,  slide  under  the  action  of 
its  own  weight;  but,  owing  to  friction,  it  will  roll.  Required  the 
circumstances  of  the    motion. 


254 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


If  the   sphere    move  from  rest  with  no    initial   impulse,  the   centre 


will  describe  a  straight  line 
parallel  to  the  element  of 
steepest  descent  Take  the 
plane  x  z,  to  contain  this 
element,  the  axis  z  vertical 
and  positive    upwards. 

The  equation  of  the  path 
will   be, 


t 


L  =  z  4*  *  tan  a  —  h  ■=.  0 ; 


whence, 


dL 

dz 


=  i; 


dL 
dx 


=  tan  a. 


The  extraneous  forces  are  the  weight  of  the  sphere  and  the  fric- 
tion.  Denote  the  first  by  W,  and  the  second  by  F.  The  nature 
of  friction  and  its  mode  of  action  will  be  explained  in  the  proper 
place,  §  354 ;  it  will  be  sufficient  here  to  say  that  for  the  same 
weight  of  the  sphere  and  inclination  of  the  plane,  it  will  be  a  con- 
stant force  acting  up  the  plane  and  opposed  to  the  motion.  We 
shall    therefore   have 


Z  =  —  Mg  -f  Fsm  a  ;     X  =  —  Fcosa, 


which    values,  and   those   above   substituted    in    Equation  (330),  give 

■■'■''■  cPx         /  d2z\ 

—  Fc.osa  —  M •  -— -  -f  (  M  a  —  .Fsin  a  +  M*  — -  I  ian  a  =  0. 

dt2         \     J  dt2/ 

;r!     But  from    the   equation  of  the   path,  we   have 

h 

d?z  =  —  d2  x  •  tan  a  ; 

»nd    eliminating   <Px   by  means  of  this   relation,  there    will    result 

i<  d*z  (F  \ 

-—-  =  sin  a  1  -aT.   —  g  sin  a  )  • 
dt2  ^M         v  ' 


MECHANICS     OF     SOLIDS.  255 

Multiplying    by    2rfz,    integrating    and    making    the    velocity     zero 
when  z  =  A,  we   have 

—  =,  F2  =  2  sin  a  {-  -  g  gin  a)  ■  (*  -  A). 
This   gives 

1  .-/  2 


C^TTZ  _„o!„jt    AT? 


I      .        {F  .       \ 

W  2  sin  «  I  —    —  ^  sin  a) 


and    bv  integration,  the    time   being  zero  when  z  =  /i, 

A  —  z  =  %  sin  a  (g  •  sin  « —   )  •  /2.       .        ,        .      («). 

Again,  all  axes  in  the  sphere  through  its  centre,  are  principal 
axes;  the  sphere  will  only  rotate  about  the  movable  axis  y,  in 
which  case  v,  and  v,  will  each  be  zero,  and  Equations  (202)  will  give 


wherein, 


B  = 

Mk?-y 

dvv 
dt 

d?-\, 
~    dt2 

r 

being 

the 

radius 

of  the 

sphere. 

Whence, 

• 

rf2  + 

Ft 

dt2 

Mk? 

Mt  =  Fr\ 

4  * 


Multiplying    by    2d^,  integrating,  and    making    the    angular  velocity 
and  the   arc  4,  vanish   together, 

dt*    ~~  Mi}       ' 
whence, 


IMk?      d\  . 

d  t    —    K  / —  •  — 7=5- ' 

V  2Fr     a/T 


256  ELEMENTS     OF     ANALYTICAL    MECHANICS 

and  by    integration,  making  t  and  -\>  vanish  together, 

F  r 
*        *  M-k* 

Also,    because    the    length  of  path   described  in    the    direction    of  the 
$)lane  is  r.-vLj  we  have,  in  addition, 

h  —  z  =  r  .  4>  •  sin  a ; 

and    eliminating  4-  from    this    and    the    above    equation,    there    will 
result 

t  ■  «/_,** *£    (A  -  «), (J) 

V  -r  .  r2.sma  v  '  w 

Dividing    Equation   (a)   by    Equation    (/>),   and  solving   with    respect 

to  /; 

^=  ^r  .^/if/   _____;       .......       .(c) 

and  this  in    Equation    (&),    gives 

f  =  Ji(h-z)    t*  +  r» 

V    */  •  sin2</,  r2  ' 

If    the    sphere    be    homogeneous,    then    will 

<2{h  -z) 
9 
it'    the    matter    be    all    concentrated    into    the    surface,  then    will 


k\  =  f  ,•  and  i  =  x/?^  •  J[i 


V  =  2  ,2  aild  ,  =  i/*l"\-    *)  •  W^  ; 

r         ■  V   # .  sin2 «     V3 

which    times   are    to    one   another    as    -y/^T  to    yl^. 

CONSTRAINED    MOTION    ABOUT   A   FIXED   POINT. 

§  227.— If  a  body  be  retained  by  a  yfoec?  point,  the  fixed  and 
what  has  been  thus  far  regarded  as  a  movable  origin  may  both  be 
taken  at  this  point;  in  which  case,  8x„  Si/.,  6z^  in  Equation  (40), 
will  be  zero,  the  first  three   terms  of  that  general    equation    ^f  equi 


MECHANICS    OF    SOLIDS.  257 

librium  will  reduce  tc  zero  independently  of  the  forces,  and  the  equi- 
librium will  be  satisfied  by    simply  making 


,w           r>                     \         *.        x  .d2  y  —  y  dl  x        _     "l 
2  P  (x  cos  ft  —  y  cos  a)  —  2  m AA   =  0 ; 


2  P  (z  cos  a  —  x  cos  7)  —2m-  — —— =  0 


dt2 

z . 

d2 

x  —  2 

'd2z 

dt2 

y 

,d2 

z  —  z . 

d2  y 

►  . . (365 


2  P  (y  cos  7  —  z  cos  /3)  —  2  m  •  £- — £  =  0  ; 


(365) 


the   accents  being    omitted    because    the    elements    ?/<,  m/,  &c,    being 
referred   to    the   same   origin,  x\  y\  z'  will  become  x,  y,  z. 

The  motion  of  the  body  about  the  fixed  point  might  be  discussed 
both  for  the  cases  of  incessant  and  of  impulsive  forces,  but  the  dtecux 
sion  being  in  all  respects  similar  to  that  relating  to  the  motion  about 
the  centre  of  inertia,  §  127  and   §  173,  we  pass  to 

CONSTRAINED    MOTION    ABOUT    A    FIXED    AXIS. 

S22S. — If  the  bodv  be  constrained  to  turn  about  a  fixed  axl*, 
both  origins  may  be  taken  upon,  and  the  co-ordinate  axis  y  t<» 
coincide  with  this  axis;  in  which  case  8x^  Sy^  $ps1  '$$  and  8u, 
in  Equation  (40),  will  be  zero,  and  to  satisfy  the  conditions  of 
equilibrium,  it  will  only  be  necessary  for  the  forces  to  fulfil  thi? 
condition, 

z  d  x  x  '  d2  z 

2  P  (z  cos  a  —  x  COS7)  —2m —f^ =  <>    '  '  (366) 

the    accents  being   omitted  for  reasons  just  stated. 

§229. — The  only  possible    motion   being   that   of    rotation,    let   us 
transform   the   above    equation    so   as  to  contain  angular  co-ordinates. 
For  this   purpose  we   have,  Equations  (36), 

x'  =  r"  sin  -\, ;     z'  =  r"cos-^  .....     (367) 

in  which  r"  denotes  the   distance   of  the  element  m  from  the  a„xis  »*. 
Omitting  the  accents,  differentiating  and  dividing    by  d  t,  we  -have' 

dx  d\       dz  c?+  ftN 

__  =  rC0s  +  _;     _=_-sm  +  ._.     •     .(368, 


258  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Now, 

Z'd2  x        x>  d2z 


1      ,  /      dx  dz\ 

it        \      dt  dt/ ' 


dfl  dt> 

whence  by  substitution,  Equations  (367)  and  (368), 

(Px  d2z  1       . /.    rf  +  \  d*± 


-  -  x  . 


tf  <2  tf  (3  £/  ^ 


V      dt/  -        dt2' 


d?-l> 
and   since    —~    must  be  the  same  for  every  element,  we  have,  Equa 
dt* 

tion    (366), 

2  m  r2  •  -yj?  =  2  P  (z  cos  a  —  ar  cos  y), 


and 

rf2  4/         2  P  •  (z  cos  a  —  or  cos  7) 
rf  t2  2  m  r2 


(369) 


That  is  to  say,  the  angular  acceleration  of  a  body  retained  by  a 
fixed  axis,  and  acted  upon  by  incessant  forces,  is  equal  to  the 
cnoment  of  the  impressed  forces  divided  by  the  moment  _>f  inertia 
with,  reference   to  this  axis. 

Denoting  the  angular  velocity  by  Vx ,  and  the  moment  of  inertia 
by  /,  we  find,  by  multiplying  Equation  (369)  by  2  c?  4*  an(^  integrating, 

IVX2  =  2^2  P(z  cos  a  —  xoosy)d-^  -f  C, 

and   supposing   the   initial  angular   velocity  to  be   F/,  we  have 

I(V2  -   F/s)  =  2fzP(zcosa  -  xcos7)d^. 

But  the  second  member  is,  §  107,  twice  the  quantity  of  work 
about  the  fixed  axis ;  whence  the  quantity  of  work  performed  be- 
tween the  two  instants  at  which  the  b<  5y  has  any  two  angular 
velocities,  is  equal  to  half  the  difference  of  the  squares  of  these 
velocities  into  the  moment  '  of  inertia,  01  to  half  the  lVng  force 
gained    or   lost    in    the    interval. 


MECHANICS    OF    SOLIDS. 


259 


Now,  /=  Mk2  =  Mt.  (I)2  =  Mt ;  so  that,  the  moment  of  inertia 
measures  that  mass  which  would,  if  concentrated  on  the  arc  \,  have 
a  living  force  equal  to  that  of  the  body  which  actually  rotates. 


COMPOUND    PENDULUM. 

§  230. — Any    body    suspended  from    a   horizontal    axis  A  B,    about 
■which    it   may    swing  with  freedom   under   the 
action   of  its  own  weight,  is  called  a  compound 
pendulum. 

The  elements  of  the  pendulum   being  acted 
upon    only  by  their  own  weights,  we  have 

P  =  mg  ;     F* "=  mf  g,  &c.  ; 

the  axis  of  z  being  taken  vertical  and  positive 
downwards, 

cos  a  =  cos  a'  =  &c.  =  0  ; 


cos  * 


cos/'  ==  &c. 


and  Equation  (369)  becomes 

d2\  2mx 


dt2 


=  -  9' 


Lmr2 


(370) 


Denote  by  e,  the  distance  A  G,  of  the  centre  of  gravity  from  the 
axis;    by  -^,    the   angle  HAG,  which 
A  G  makes  with  the  plane  yz\   by  xn 
the    distance  of  the   centre  of  gravity 
from  this  plane ;    then  will 

x4  =  e  .  sin  -^  ; 

and  from  the   principles  of  the  centre 
of  gravity, 

2  m  x  =  Mxt  =  M.  e .  sin  ^ ; 
which  substitute!  above,  gives 


d2\ 


=  ~  9 


M .  e .  sin  <\t 
2mr2 


(371) 


260  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Multiplying  by  2d-],,  and  integrating, 

d\2  M.e. 

d  t2  2mr* 

Denoting  the  initial  value  of  >L  by  a,  we  have 

_.         Me    ,  _, 

0  =  29'~ r-coset  -f  C; 


whence, 


d-l2        n        M.e     .        . 


but 


C°S+  =  1-0-f  172^4-^ 

a2  a* 

cos  a  =  1 4- &c. 

1.2  x  1.2-3.4 

and  taking  the  value  of  4*?  so    small    that  its   fourth    power   may  dp 
neglected  in  comparison  with  radius,  we  have 

cos  4/  —  cos  a  ss  -— —  ; 

2 

which  substituted  above,  gives,  after  a  slight  reduction,  and  replacing 
2mr2  by  its  value  given  in  Equation  (216), 

d  -vf- 


V      e.g  I 


v/-S 

ihe  negative  sign  being  taken    because  \  is  a    decreasing  function  of 
the  time. 

Integrating,  we  have 

/Jc  2  _L  e2  _j  r 

t  =  \/'        —  -cos      — (373) 

V        e.^  a  v 

The   constant   of    integration  is   zero,  because  when  -^  =  a,  we    have 
i  =  0. 


MECHANICS    OF    SOLIDS.  261 

Making  -\.  =  —  a,  we  have 

Ik2  +  e2 

which  gives  the  time  of  one  entire  oscillat'on,  and  from  which  we 
conclude  that  the  oscillations  of  the  same  pendulum  will  be  isochro- 
nal, no  matter  what  the  lengths  of  the  arcs  of  vibration,  provided 
they  be  small. 

If  the  number  of  oscillations  performed  in  a  given  interval,  say 
ten  or  twenty  minutes,  be  counted,  the  duration  of  a  single  oscillation 
will  be  found  by  dividing  the  whole  interval  by  this  number. 

Thus,  let  6  denote  the  time  of  observation,  and  JV  the  number  of 
oscillations,  then  will 


=i=*v 


k2  +  e2 
e.g 


and  if  the  same  pendulum  be  made  to  oscillate  at  some  other  location 
during  the  same  interval  &,  the  force  of  gravity  being  different,  the 
number  N'  of  oscillations  will  be  different ;  but  we  shall  have,  as 
before,  g'  being  the  new  force  of  gravity, 

6     _  A2  +  <?2~ 

N'  ~  *     V       e.g' 

Squaring   and  dividing  the  first  by  the  second,  we  find 

N'2        g' 


N2   ~   g 


(374)' 


that  is  to  say,  the  intensities  of  the  force  of  gravity,  at  different 
places,  are  to  each  other  as  the  squares  of  the  number  of  oscilla- 
tions performed  in  the  same  time,  by  the  same  pendulum.  Hence, 
if  the  intensity  of  gravity  at  one  station  be  known,  it  will  be  easy 
to  find  it   at   others. 

§  231. — From  Equation    (372),  we  have 

-j-j  .  2  m  r2  =  2  M .  g  .  e  (cc  s  >L  —  cos  a)  ;     .     -     (375) 


262  ELEMENTS    OF    ANALYTICAL    MECHANICS, 

and  making 

-~1  =   F, :    2  m  **2  =  /:    e  (cos  -L  —  cos  a)  =  H ; 

we  have 

I.V*  =  2M.g.H; (376) 

in  which  H,  denotes  the  vertical  height  passed  over  by  the  centre 
of  gravity,  and  from  which  it  appears  that  the  pendulum  will  come 
to  rest  whenever  -^  becomes  equal  to  a,  on  either  side  of  the  ver- 
tical  plant   through   the   axis. 

§  232. — If  the  whole  mass  of  the  pendulum  be  conceived  to  be 
concentrated  into  a  single  point,  the  centre  of  gravity  must  go 
there  also,  and  if  this  point  be  connected  with  the  axis  by  a  medium 
without  weight  and  inertia,  it  becomes  a  simple  pendulum.  Deno- 
ting the  distance  of  the  point  of  concentration  from  the  axis  by  /, 
we  have 

kt  =  0 ;     e  =  Z, 

which  reduces  Equation  (374)  to 

t  =  «-\P- (377) 

If  the  point    be  so  chosen  that 


I  Ik?  +j? 

a    -\"~e~:Y~; 


9 
or. 


•        •        • 


(378) 


the  simple  and  compound  pendulum  will  perform  their  oscillations  in 
the  same  time.  The  former  is  then  called  the  equivalent  simple  pen- 
dulum;  and  the  point  of  the  compound  pendulum  into  which  the 
mass  may  bf  concentrated  tc  satisfy  this  condition  of  equal  duration, 
is  called  the  centre  of  oscillation.  A  line  through  the  centre  of 
oscillation  and  parallel  to  the  axis  of  suspension,  is  called  an  axis  of 
oscillation. 


MECHANICS    OF    SOLIDS.  26o 

£233. — The    axes  of   oscillation    and  of   suspension  are    reciprocal 
Denote  the  length  of  the  equivalent  simple  pendulum  when   the  com 
pound  pendulum  is  inverted  and  suspended  from  its  axis  of  oscillation, 
by  V ,  and  the  distance  of  this  latter  axis  from  the  centre   of  gravity 
by  e[  then  will 

/   =  e  4-  «'     or     e'  =  /  —  e\ 

and,  Equation  (378), 

9  _  *,2  ±  e'2    _  *,2  +  (/  -  «)» 
1    ~  ?  ~  l-e 

and  replacing  /,  by  its  value  in  Equation  (378),  we  find 

h  2  4-  e2 
e 

That  is,  if  the  old  axis  of  oscillation  be  taken  as  a  new  axis  of  su;» 
pension,  the  old  axis  of  suspension  becomes  the  new  axis  of  oscilla- 
tion. This  furnishes  an  easy  method  for  finding  the  length  of  an 
equivalent  simple   pendulum. 

Differentiating    Equation  (378),  regarding  /  and  e   as    variable,    we 
have 

ii  ~  g2  - k? 

de  e2 

and  if  /   be   a   minimum, 

d-  =  o  =  e2  ~  k<2  - 


de  e2 

whence, 

e  •=.  kt. 

But  when  /  is  a  minimum,  then  will  t  be  a  minimum,  Equa- 
tion (377).  That  is  to  say,  the  time  of  oscillation  will  be  a 
minimum  when  the  axis  of  suspension  passes  through  the  principal 
centre  of  gyration,  and  the  time  will  '  e  longer  in  proportion  as  the 
axis  recedes  from    that   centre. 


264 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


u 


m 

i 
1 

f 
i 

i 

I 

- 

* 

M    ] 
— i L 

i 

i 

Let  A  and  a£*  be  twc  acute  parallel  prismatic  axes  firmly  con 
nected  with  the  pendulum,  the  aaute  edges 
being  turned  towards  each  other.  The 
oscillation  may  be  made  to  take  place 
about  either  axis  by  simply  inverting  the 
pendulum.  Also,  let  if  be  a  sliding  mass 
capable  of  being  retained  in  any  position 
by  the  clamp-screw  H.  For  any  assumed 
position  of  M,  let  the   principal    radius  of 

gyration    be    GC;    with    G    as    a    centre,  \°  ) 

G  C  as  radius,  describe  the  circumference 
CSS'.  From  what  has  been  explained, 
the  time  of  oscillation  about  either  axis 
will   be    shortened    as    it    approaches,    and 

lengthened  as  it  recedes  from  this  circumference,  being  a  minimum, 
or  least  possible,  when  on  it.  By  moving  the  mass  M,  the  centre 
of  gravity,  and  therefore  the  gyratory  circle  of  which  it  is  the 
centre,  may  be  thrown  towards  oither  axis.  The  pendulum  bob  being 
made  heavy,  the  centre  of  gravity  may  be  brought  so  near  one  of 
the  axes,  say  A\  as  to  place  the  latter  within  the  gyratory  cir- 
cumference, keeping  the  centre  of  this  circumference  between  the 
axes,  as  indicated  in  the  figure.  In  this  position,  it  is  obvious  that 
anv  motion  in    the    mass  M  would   at   the   same   time  either   shorten 

« 

or  lengthen  the  duration  of  the  oscillation  about  both  axes,  but 
unequally,  in  consequence  of  their  unequal  distances  from  the  gyratory 
circumference. 

The  pendulum  thus  arranged,  is  made  to  vibrate  about  each  axis 
in  succession  during  equal  intervals,  say  an  hour  or  a  day,  and  the 
number  of  oscillations  carefully  noted;  if  these  numbers  be  the 
same,  the  distance  between  the  axes  is  the  length  /,  of  the  equiva- 
lent simple  pendulum  ;  if  not,  then  the  weight  M  must  be  moved 
towards  that  axis  whose  number  is  the  least,  and  the  trial  repeated 
till  the  numbers  are  made  equal.  The  distance  between  the  axes 
may  be   measured  by  a  scale  of  equal  parts. 

§  234. — From  this  value  of  /,  we  may  easily  find  that  of  the  simple 
tecotiiTs   pendulum       that  is  to  say,  the  sirrple  pendulum  which  vvilj 


MECHANICS    OF     SOLIDS.  265 

perform  its  vibration  in  one  second.  Let  iV,  be  the  number  of 
vibrations  performed  in  one  hour  by  the  compound  pendulum  whose 
equivalent  simple  pendulum  is  /;  the  number  performed  in  the 
same  time  by  the  second's  pendulum,  whose  length  we  will  denote 
by  V,  is  of  course  3600,  being  the  number  of  seconds  in  1  hour, 
and  hence, 

N  V  g 

1*  fl 


3600'  V  g  * 


and  because  the  force  of  gravity  at  the  same  station  is  constant, 
we  find,  after  squaring  and  dividing  the  second  equation  by  the  first, 

*»-  -±£-     .......     (379) 

(3600)2  v        ' 

Such  is,  in  outline,  the  beautiful  process  by  which  Kater  determined 
the  length  of  the  simple  second's  pendulum  at  the  Tower  of  London 
10   be    39,13908  inches,  or  3,26159  feet. 

As  the  force  of  gravity  at  the  same  place  is  not  supposed  to 
change  its  intensity,  this  length  of  the  simple  second's  pendulum 
must  remain  forever  invariable ;  and,  on  this  account,  the  English 
have  adopted  it  as  the  basis  of  their  system  of  weights  and  measures. 
For  this  purpose,  it  was  simply  necessary  to  say  that  the  3-,2lTT59th 
part  of  the  simple  second's  pendulum  at  the  Tower  of  London  shall 
be  one  English  foot,  and  all  linear  dimensions  at  once  result  from 
the  relation  they  bear  to  the  foot ;  that  the  gallon  shall  contain 
YT2'SXh  °f  a  CUD,C  f°ot>  aRd  all  measures  of  volume  are  fixed  by  the 
relations  which  other  volumes  bear  to  the  gallon  ;  and  finally,  that 
a  cubic  foot  of  distilled  water  at  the  temperature  of  sixty  degrees 
Fahr.  shall  weigh  one  thousand  ounces,  and  all  weights  are  fixed  by 
the   relation  they   bear   to   the  ounce. 

§235. — It  is  now  easy  to  find  the  apparent  force  of  gravity  at 
London  ;  that  is  to  say,  the  force  of  gravity  as  affected  by  the  cen- 
trifugal force  and  the  oblateness  of  the  earth.     The  time  of  oscillation 


266  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

being    one   second,  and    the   length  of  the    simple    pendulum    3,26159 
feet,  Equation  (377)  gives 


3,26159 
—  *\/ ; 

9 

whence" 

g  =  **  (3^6159)  =  (3,1416)2.  (3,26159)  =  32,1908  feet. 

From  Equation  (377),  we    also  find,  by  making    t  one    second, 


and    assuming 


we   have 


I  =  x  -f-  y  cos  2  4s 


JL  as  #  4-  y  cos  2  4 (380) 

Now  starting  with  the  value  for  g  at  London,  and  causing  the 
sume  pendulum  to  vibrate  at  places  whose  latitudes  are  known,  we 
obtain,  from  the  relation  given  in  Equation  (374) ',  the  corresponding 
values  of  g,  or  the  force  of  gravity  at  these  places ;  and  these 
values  and  the  corresponding  latitudes  being  substituted  successively 
in  Equation  (380),  give  a  series  of  Equations  involving  but  two  un- 
known quantities,  which  may  easily  be  found  by  the  method  of 
least   squares. 

In  this    way  it   has   been  ascertained  that 

«r2.*  =  32,1808    and     «r2.y  =  -  0,0821  ; 

whence,  generally, 

t  f 

g  =  32,1808  -  0,0821  cos  2  4  ;      ....     (381) 

and   substituting   this   value    in    Equation    (377),  and    making    t  =  1, 
we  find 

/ 

I  =  3,26058  -t  0,008318  cos  2  4     •     .     .     .     (382) 

JSuch   is    the   length  of   the   simple   second's    pendulum   at   any   place 
of  which    the   latitude  is  4*. 


MECHANICS     OF     SOLIDS, 


267 


If  we    make   +  =  40°  42'  40",  the    latitude   of  the    City  Hall  of 
Np.w  York,  we   shall  find 


I 


ft- 


in. 


3,25938  =  39,11256. 

§236. — The  principles  which  have  just  been  explained,  enable  us 
to  find  the  moment  of  inertia  of  any  body  turning  about  a  fixed 
axis,  with  great  accuracy,  no  matter  what  its  figure,  density,  or  the 
distribution  of  its  matter.  If  the  axis  do  not  pass  through  its  centre 
of  gravity,  the  body  will,  when  deflected  from  its  position  of  equi- 
librium, oscillate,  and  become,  in  fact,  a  compound  pendulum  ;  and 
denoting  the  length  of  its  equivalent  simple  pendulum  by  /,  we  have, 
after   multiplying  Equation  (378)  by  M, 

M.l.e  =  M  (k*  +  e2)  =  2  m  r*  ;     .     .     .     .     (383) 
or  since 


W 


W 

M  =  — , 

9 


•  I .  e  =  Z,mr2, 


(384) 


'.n   which    W  denotes    the   weight  of  the    body. 

Knowing    the    latitude    of    the   place,  the    length    /'   of  the    simple 
second's  pendulum  is  known  from  Equation  (382)  ;    and  counting  the 
number    N   of    oscillations    performed    by     the    body    in    one    hoar 
Equation  (379)   gives 

.        V  •  (3600)2 

To  find  the  value  of  e,  which  is 
the  distance  of  the  centre  of  gravity 
from  the  axis,  attach  a  spring  or 
other  balance  to  any  point  of  the 
body,  say  its  lower  end,  and  bring 
the  centre  of  gravity  to  a  horizontal 
plane  through  the  axis,  which  posi- 
tion will  be  indicated  by  the  max- 
imum reading  of  the  balance.  De- 
noting by  a,  the  distance  from  the  axis  C  to  the  po'nt  of  support  A', 

17 


268 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


and  by  b,  the  maximum  indication  of  the  balance,  we  have,  frcm 
the  principle  of  moments, 

b  a  =  We. 

The  distance  a,  may  be  measured  by  a  scale  of  equal  parts.  Sub- 
stituting the  values  of  W,  e  and  I  in  the  expression  for  the  moment 
of   inertia,  Equation  (384),  we   get 


b.a.l'.(3600)2 

9-N2 


=  I. 


(385) 


If  the  axis  pass  through  the  centre  of  gravity,  as,  for  example, 
in  the  flu-wheel,  it  will  not  oscillate;  in  which  case,  take  Equation 
(383),  from  which  we  have 

Mk2  =  M.l.e  -  Me2. 

Mount  the  body  upon  a  parallel  axis  A,  not  passing  through  the  cen~ 
tre  of  gravity,  and  cause  it  to  vibrate 
tor  an  hour  as  before ;  from  the  num 
her  of  these  vibrations  and  the  length 
of  the  simple  second's  pendulum,  the 
value  of  I  may  be  found;  M  is  known, 
bc'ing  the  weight  W  divided  by  g  ;  and 
e  may  be  found  by  direct  measure- 
ment, or  by  the  aid  of  the  spring 
balance,  as  already  indicated;  "whence  kt  becomes  known. 


MOTION     OF   A   BODY    ABOUT   AN     AXIS    UNDER   THE   ACTION    OK     IMPUL- 
SIVE   FORCES. 


§  237. — If  the  forces  be  impulsive,  we  may,  §  170,  rep) 'ice  in 
Equation  (366)  the  second  differential  co-efficients  of  f,  y,  2,  by  the 
first  differential  co-efficients  of  the  same  variables,  which  will  reduce 
it  to 

~  tw  \        ~        Z(lx  —  xdz 

2  Piz  cos  a  —  x  cos y)  —  2  m  • -= =  0  : 

at 


MECHANICS    OF    SOLIDS. 


269 


and    replacing    dx  and  rfz,    by  their  values    in    Equations    (368),  we 
find 


d\         2  P  (z  cos  a  —  x  cos  y) 
~dT  Zmr2 


(386) 


That  is,  the  angular  velocity  of  a  body  retained  by  a  fixed  axis,  and 
subjected  to  the  simultaneous  action  of  impulsive  forces,  is  equal  to  the 
sum  of  the  moments  of  the  impressed  forces  divided  by  the  moment  of 
inertia  with  reference  to  this  axis. 


BALLISTIC    PENDULUM. 

§  238. — In  artillery,  the  initial  velocity  of  projectiles  is  ascertained 
by  means  of  the  ballistic  pendulum, 
which  consists  of  a  mass  of  matter 
suspended  from  a  horizontal  axis 
in  the  shape  of  a  knife-edge,  after 
the  manner  of  the  compoimd  pen- 
dulum. The  bob  is  either  made 
of  some  unelastic  substance,  as 
wood,  or  of  metal  provided  with 
a  large  cavity  filled  with  some- 
soft  matter,  as  dirt,  which  re- 
ceives the  projectile  and  retains 
the  shape  impressed  upon  it  by  the 
blow 

Denote  by  V  and  m,  the  initial  velocity  and  mass  of  the  ball ; 
Vl  the  angular  velocity  of  the  ballistic  pendulum  the  instant  after 
the  blow,  /  and  M  its  moment  of  inertia  and  mass.  Also  let  I 
represent  the  distance  of  the  centre  of  oscillation  of  the  pendulum 
from  the  axis  A.  That  no  motion  may  be  lost  by  the  resistance 
of  the  axis  arising  from  a  shock,  the  ball  must  be  received  in  the 
direction  of  a  line  passing  through  this  centre  and  perpendicular  to  the 
plane  of  the  axis  and  line  A  0.     With  this  condition,  Eq    (386)  give? 

4* 


dt 


Vx 


m.V.l 


mV 


TLmr1         iJft4-«NJi**Hh**J        i  Jf +»»).« 


270  ELEMENTS   OF   ANALYTICAL   MECHANICS, 

whence 

m 

and  supposing  the  angular  velocity  communicated  to  the  pendulum  tc 
be  equal  to  that  acquired  by  falling  from  rest  through  the  initial  arc 
a,  in  Equation  (372),  we  have,  from  that  equation  and  Equation  (216), 
by  writing  e  for  d, 


and  Eq.   (374), 


-  =  V 


9e 


which  substituted  above  ogives 


V i  =  2  -  •  sin  £  a ; 


and  this  in  the  value  for  V  gives,  afier  substituting  for  the  ratio  of  the 
masses  that  of  their  weights, 

W  -+-  w    <x  .     . 

V=2 e  •  sin  A  a (387. 

w  t  2  v        ' 

From  this  equation  we  may  find  the  initial  velocity  V ';  and  for 
this  purpose,  it  will  only  be  necessary  to  have  the  duration  of  a  single 
oscillation,  and  the  amplitude  of  the  arc  described  by  the  centre  of 
gravity  of  the  pendulum.  The  process  for  finding  the  time  has  been 
explained.  To  fiud  the  arc,  it  will  be  sufficient  to  attach  to  the 
Kiwer  extremity  of  the  pendulum  a  pointer,  and  to  fix,  on  a  permanent 
stand  below,  a  circular  graduated  groove,  whose  centre  of  curvature  is 
at  A  ;  the  groove  being  filled  with  some  soft  substance,  as  tallow,  the 
pointer  will  mark  on  it  the  extent  of  the  oscillation.  Knowing  thus 
the  arc  a,  and  the  value  of  ey  found  as  already  described,  §  236  we 
tiave  V. 


MECHANICS    OF    SOLIDS.  271 


THE     GUN     PENDULUM. 

This  consists  of  a  gun  suspended  from  a  horizontal  axis.  The  shot 
is  fired  from  the  gun,  and  its  velocity  is  inferred  from  the  recoil,  as 
in  the  Ballistic  Pendulum.  The  forces  measured  by  the  quantities  of 
motion  developed  by  the  expansive  action  of  the  exploded  powder, 
must  be  in  equilibrio.     Make 

V  =  velocitv  of  the   ball  on    leaving  the  gun, 

I 

nV  =  average  velocity  of  the  inflamed  powder, 
Vf  —  angular  velocity  of  pendulum  on  parting  from  shot, 
W  =  weight  of  gun   pendulum, 
Wb  —        "  hall   and   wad, 

We  =        "  the  charge  of  powder  and   bag, 

Wp  —        "  "  tt         of  powder  alone, 

6  =  diameter  of  bore, 

d  =  diameter  of  ball, 

e  =  distance  of  axis  of  bore  from  axis  of  suspension. 

•> 

The  quantity  of  motion  in  ball  and  wad,  on  leaving  the  gun,  will  be 
Y\    the    corresponding    pressure    on    the    bottom    of  the    gun    is  tc 

AS 

that  which  generates  this  motion,  as  the  area  of  a  cross-section  of  th«. 
bore  is  to  that  of  a  great  circle  of  the  ball.  Again,  the  blast  of  the 
powder  will  continue  its  action  on  the  gun  after  the  ball  leaves  it. 
Let  this  action  be  proportional  to  the  charge  of  powder.  The  moment 
of  the  force  impressed  upon  the  pendulum,  in  reference  to  the  axis  of 
suspension,  will  be  given  by  Eqs.  (384)  and  (229)  ;  and  taking  the 
moments  of  the  other  forces  in  reference  to  the  same  axis,   we  have 

9  9  «*        9  9 

tX\  which  n\  like  rc,  is  a  constant  to  be  determined  by  experiment: 
and  from   which  we  find 

IV  .V.l.e 


r= 


* 


W.-t-  -  +  nW  .*  +  n'Wm.€ 


272  ELEMENTS    OF    ANALYTICAL    MECHANICS. 


^n 


The    living    force    with    which    the    pendulum    separates   from     the    ball 

must  equal  twice  the  work  performed  by  the   weight  while  the  centre 

of  gravity  is  moving  to  the  highest  point;    whence 

W 

yi f .  /  .  e  —  2  W  .  e  .  versine  a  =  4  W , .  e  .  sin2  i  a, 

9  f  g 

fn    which    a    denotes    the    greatest    inclination     of    e    to    the    vertical. 
Whence 

which  substituted  above  gives, 

Wt'^  +  nW^n'W 


*      £ 

V= j- sin  ^a       ....     ifc88). 


p 


The  methods  for  finding  e  and  a  are  the  same  as  in  the  ballistic 
pendulum.  To  find  n  and  w',  fire  the  ball  from  the  gun  into  the 
ballistic  pendulum ;  the  effect  upon  the  latter  will  give  the  initial 
velocity  V.  Repeat  as  often  as  may  be  thought  desirable,  and  with 
different  charges.  The  corresponding  initial  velocities  substituted  in 
Eq.  (388),  will  give  as  many  equations  as  trials.  These  equations  will 
contain  only  n  and  n'  as  unknown  quantities,  which  may  be  found 
by  the  method  of  least  squares.  For  full  and  valuable  information 
on  this  subject,  consult  Mordecai's  "Experiments  on  Gunpowder." 


PART    II. 


MECHANICS  OF  FLUIDS. 


INTRODUCTORY    REMARKS. 

§  239. — The  physical  condition  of  every  body  depends  upon  the  rela- 
tions subsisting  among  its  molecular  forces.  In  the  vast  range  of  relations, 
from  those  which  distinguish  a  solid  to  those  which  determine  a  gas  or 
vapor,  bodies  are  found  in  all  possible  conditions — solids  run  impercepti- 
bly into  liquids,  and  liquids  into  vapors  or  gases.  Hence  all  classification" 
of  bodies  founded  on  their  physical  properties  alone,  must,  of  necessity, 
be  arbitrary. 

§  240. — Any  body  whose  elementary  particles  admit  of  motion  among 
each  other  is  called  a  fluid — such  as  water,  wine,  mercury,  the  air,  and, 
in  general,  liquids  and  gases;  all  of  which  are  distinguished  from  solids 
by  the  great  mobility  of  their  particles  among  themselves.  This  distin- 
guishing property  exists  in  different  degrees  in  different  liquids — it  is 
greatest  in  the  ethers  and  alcohol ;  it  is  less  in  water  and  wine;  it  is  still 
less  in  the  oils,  the  sirups,  greases,  and  melted  metals,  that  flow  with 
difficulty,  and  rope  when  poured  into  the  air.  Sueh  fluids  arc  said  to  be 
viscous,  or  to  possess  viscosity.  Finally,  a  body  may  approach  so  closely 
both  a  solid  and  liquid,  as  to  make  it  difficult  to  assign  it  a  place  among 
either  class,  as  paste, putty,  sealing  wax,  and  the  like. 


27i       ELEMENTS    OF    ANALYTICAL    MECHANICS. 

§  241. — Fluids  arc  divided  in  mechanics  into  two  classes,  viz. :  com- 
pressible and  incompressible.  The  term  incompressible  cannot,  in  strict- 
ness of  propriety,  be  applied  to  any  body  in  nature,  all  being  more  or  less 
compressible ;  but  the  enormous  power  required  to  change,  in  any  sensi- 
ble degree,  the  volumes  of  liquids,  seems  to  justify  the  term,  when  applied 
to  them  in  a  restricted  sense.  The  gases  are  highly  compressible.  All 
liquids  will,  therefore,  be  regarded  as  incompressible ;  the  gases  as  com- 
pressible. 

g  242. — The  most  important  and  remarkable  of  the  gaseous  bodies 
is  the  atmosphere.  It  envelops  the  entire  earth,  reaches  far  beyond  the 
tops  of  our  highest  mountains,  and  pervades  every  depth  from  which  it 
is  not  excluded  by  the  presence  of  solids  or  liquids.  It  is  even  found  in 
the  pores  of  these  latter  bodies.  It  plays  a  most  important  part  in  all 
natural  phenomena,  and  is  ever  at  work  to  influence  the  motions  within 
it.  It  is  essentially  composed  of  oxygen  and  nitrogen,  in  a  state  of 
mechanical  mixture.  The  former  is  a  supporter  of  combustion,  and,  with 
the  various  forms  of  carbon,  is  one  of  the  principal  agents  employed  in 
the  development  of  mechanical  power. 

The  existence  of  gases  is  proved  by  a  multitude  of  facts.  Contained 
in  an  inflexible  and  impermeable  envelope,  they  resist  pressure  like  solid 
bodies.  Gas,  in  an  inverted  glass  vessel  plunged  into  water,  will  not  yield 
its  place  to  the  liquid,  unless  some  avenue  of  escape  be  provided  for  it. 
Tornadoes  which  uproot  trees,  overturn  houses,  and  devastate  entire  dis- 
tricts, are  but  air  in  motion.  Air  opposes,  by  its  inertia,  the  motion  of 
other  bodies  through  it,  and  this  opposition  is  called  its  resistance. 
Finally,  we  know  that  wind  is  employed  as  a  motor  to  turn  mills  and  to 
give  motion  to  ships  of  the  largest  kind. 

g  243. — In  the  discussions  which  are  to  follow,  fluids  will  be  consid- 
ered as  without  viscosity ;  that  is  to  say,  the  particles  will  be  supposed 
to  have  the  utmost  freedom  of  motion  among  each  other.  Sucli  fluids 
are  said  to  be  'perfect.  The  results  deduced  upon  the  hypothesis  of  per- 
fect fluidity  will,  of  course,  require  modification  when  applied  to  fluids 
possessing  sensible  viscosity.  The  nature  and  extent  of  these  modifica- 
tions can  be  known  only  from  expeiiments. 


MECHANICS    OF    FLUIDS.  275 


mariotte's    law. 


§  244. — Gases  readily  contract  into  smaller  volumes  when  pressed  ex- 
ternally ;  they  as  readily  expand  and  regain  their  former  dimensions  when 
the  pressure  is  removed.     They  arc  therefore  both  compressible  and  elastic. 

It  is  found  by  experiment  that  the  change  in  volume  is,  for  a  constant 
temperature,  very  nearly  proportional  to  the  change  of  pressure.  The 
density  of  the  same  body  is  inversely  proportional  to  the  volume  it  occu- 
pies. If,  therefore,  P  denote  the  pressure  upon  a  unit  of  surface  which 
will  produce,  at  a  given  temperature,  say  0°  Centr.,  a  density  equal  to 
unity,  and  D  any  other  density,  and  p  the  pressure  upon  a  unit  of  surface 
which  will,  at  the  same  temperature  of  the  gas,  produce  this  density, 
then,  according  to  the  experiments  above  referred  to,  will 

• 

p  =  P  .  D .     .     (389) 

This  law  was  investigated  by  Boyle  and  Mariotte,  and  is  known  as 
Mariotte's  Law.  It  has  been  found  to  be  very  nearly  true  for  all  gases 
which  are  not  liquefied  when  subjected  to  great  pressure  and  cold,  and 
which  are  therefore  called  permanent  gases. 


LAW     OF     THE     PRESSURE,     DENSITY,     AND     TEMPERATURE. 

§  245. — Under  a  constant  pressure,  all  bodies  are  expanded  by  heat ; 
under  a  constant  volume,  their  elastic  force  is  increased  by  the  same 
agent.  Experiment  has  shown  that  the  laws  of  these  changes  for  perma- 
nent gases  may  be  expressed  by 

p  =  P.B.(l  +  «0); (390) 

in  which  p  denotes  the  pressure  upon  a  unit  of  surface,  D  the  density  of 
the  gas,  0  the  difference  between  the  actual  and  some  standard  tempera- 
ture, and  a  a  constant  which  is  equal  to  y\s  —  0,003665  when  the 
standard  is  0°  centr.,  and  6  is  expressed  in  units  of  that  scale. 

First  supposing  D  and  6  variable  and  p  constant;  then  p  and  0 
variable  and  D  constant,  Equation  (390)  gives 

dD  a.D  dp  ap  ii 

idz  ~rr^;    de~  i  +  ao w 


276      ELEMENTS    OF    ANALYTICAL    MECHANICS. 

The  quantity  of  heat,  denoted  by  q,  necessary  to  change  the  temperature 
0  degrees  from  the  assumed  standard,  will  be  a  function  of  $>,  D,  6 ;  but 
because  of  Equation  (390)  we  may  write 

q=f(V,p) (b) 

The  increment  of  heat  which  will  raise  a  body's  temperature  one  degree, 
is  called  its  specific  heat.  The  specific  heat  being  the  increment  of  q  for 
each  unit  of  0,  if  c  denote  the  specific  heat  when  the  pressure  is  constant, 
and  ct  that  when  the  density  is  constant,  then  will, 

dq        dq    dD 
C  =  dd~dD'~dd' 


or,  Equations  (a), 


dq       dq   dp 
C,  =  db  =  dp'~dd" 


dq       a.D 


dD'l  +  ad' 
dq       a.p 


and  by  division,  making  c  =  y .  c  , 

in  which  y  denotes  the  ratio  of  the  specific  heat  of  the  gas  at  a  constant 
pressure  to  that  at  a  constant  density.  This  ratio  is  known  from  experi- 
ment to  be  constant  for  atmospheric  air  and  other  permanent  gases.  The 
experiments  of  Cazin  make  its  value  1.41  for  all  permanent  gases,  and 
those  of  Dulong  on  perfectly  dry  air  1,417.  Regarding  y  as  constant, 
the  integration  of  the  foregoing  equation  gives 


Jii 


I /  (See  Appendix  No.  S.) 

D 


in  which  /  denotes  any  arbitrary  function  of  the  quantity  within  the 
parenthesis,  and  from  which,  denoting  the  inverse  functions  by  F,  we 
may  write, 

p  =  Dr.F(q) (c) 


MECHANICS    OF    FLUIDS.  277 

From  Equation  (390)  we  have, 

o=      I      --  =  --.i)y-KF(q)^-.    .    .    (d) 

a.P.D       a       a.P  yif       a  v  ' 

Sudden  compression  increases  and  a  sudden  expansion  decreases  the 
temperature  of  bodies,  and  if  q  remain  the  same,  while  suddenly 
p,  D,  0,  become  p\  I) ',  0',  we  have, 

p'  =  D*  .F(q) .(e) 


1     „,-.   -  1 

«.  P  a 


<r  =  -!-=&". FM-t.    ......    (g) 


Eliminating  F(q)  first  from  Equations  (c)  and  (e),  and  then  from 
Equations  (d)  and  (g),  we  have,  replacing  y  and  a  by  their  numerical 
values, 


p'=p( 


nyurn 


/J)'\  0.4H 

6'  =  (273  +  0)  {—)      —  273 (392) 

These  equations  give  the  relation  between  the  densities,  elastic 
forces,  and  the  temperatures  of  a  gas  suddenly  compressed  or  dilated, 
and  retaining  the  quantity  of  its  heat  unchanged. 

The  pressure  being  constant,  make,  in  Equation  (390), 

0  =  0,  D  =  D  . 

and  divide  same  equation  by  the  result;  we  find 

Make  p  =  Dm.hn.g'  =  weight  of  a  column  of  mercury  at  standard 
temperature  Fahr.  T,  and  resting  on  a  base  unity,  in  Lat.  45°,  where 
gravity  is  g'.  These  in  Equation  (389)  give,  after  writing  0,00204  for  «, 
and    t°  —  32°    for  0, 

p  =  D™-K-J_    j-x  +  ^o  _  32o)    0?00204].    .     .     (393) 
If   the    temperature    of   the    mercury    vary    from    the    standard    Tt 


278 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


and  become    T'  then  will  Dm  also  vary  and  become  D'm  and  to  exert 
the    same  pressure    htl    must   have    a    new     value     A,    and    such  that 

D*  •  K    •  9'    =    D'n    •   ^  •  9'. 

Mercury    expands    or    contracts    0,0001001th    part    of    its    entire    vol 
ume    for    each    degree   of  Fahr.    by    which    it   increases  or    diminishes 
its    temperature.        And     as    the    density    of    the    same    body    varies 
inversely    as    its    volume,    we    have 

4E  =  A,J1  4-  (T-  7").  0,0001001] 

which    substituted    above    gives 

.     .        h„=h[\  +  (T-  T).  0,0001001] (3<H) 

EQUAL     TRANSMISSION     OF     PRESSURE. 


§246. — Let  EHL,  represent  a  closed  vessel  of  any  shape,  with 
which  two  piston  tubes  A  B'  and 
D  C  communicate,  each  tube  be- 
ing provided  with  a  piston  that 
fits  it  accurately  and  which  may 
move  within  it  with  the  utmost 
freedom.  The  vessel  being  filled 
with  any  fluid,  let  forces  P  and 
P\  be  applied,  the  former  per- 
pendicularly to  the  piston  A  B, 
and  the  latter  in  like  direction 
to  the  piston  CD,  and  suppose 
these    forces    in    equilibrio,    which 

they  may  be,  since  the  fluid  cannot  escape.  Now  let  the  piston 
A  B  be  moved  to  the  position  A'  B' ;  the  piston  CD  will  take 
some  new  position,  as  CD'.  And  denoting  by  s  and  »',  the  dis- 
tances A  A'  and  C  C,  respectively,  we  have,  from  the  principle  of 
virtual  velocities, 

Ps  =  P'Y. 

Denote  the  area  of  the  piston  A  B  by  a,  and  that  of  the  piston 
CD  by  a',  then  will  the  volume  of  the  fluid  which  was  thrust  from 
the    tube  A  B\  be   measured   by   a  .  9,  and  that  which  entered  the  tut>e 


MECHANICS    OF    FL7IDS  279 

0  C,  will  be  measured  by  a' s'.  But  the  jressuie  upon  the  pistons 
and  the  temperature  remaining  the  same,  the  entire  volume  of  the 
fluid    in    th«    vessel    and    tubes    will    be   unchanged.       Hence, 

as  —  a'  s'  \ 

dividing  the  equation  above    by  this  one,  we    have 

P         P' 

-  =  - 396) 

a  a 

That  is  to  say,  two  forces  applied  to  pistons  which  communicate  freely 
with  each  other  through  the  intervention  of  some  confined  fluid,  will 
be  in  equilibrio  when  their  intensities  are  directly  proportional  to  the 
areas    of  the  pistons  upon  which  they  act. 

This  result  is  wholly  independent  of  the  relative  dimensions  and 
positions  of  the  pistons ;  and  hence  we  conclude  that  any  pressure 
communicated  to  one  or  more  elements  of  a  fluid  mass  in  equilibrio,  is 
equally  transmitted  throughout  the  whole  fluid  in  every  direction.  This 
law  which  is  fully  confirmed  by  experiment,  is  known  as  the  prin- 
ciple of  equal  transmission  of  pressure. 

§247. — Let  a  become  the  superficial  unit,  say  a  square  inch  or 
square  foot,  then  will  P  be  the  pressure  applied  to  a  unit  of  sur- 
face, and,  Equation   (396), 

Pf  =  P  a'. (397) 

That  is,  the  pressure  transmitted  to  any  portion  of  the  surface  of 
the  containing  vessel,  will  be  equal  to  that  applied  to  the  unit  of 
surface  multiplied  by  the  area  of  the  surface  to  which  the  transmis- 
sion   is   made. 

§  248 — Since  the  elements  of  the  fluid  are  supposed   in  equilibrio, 
the   pressure  transmitted   to  the   surface  through   t'le   elements   in  con- 
tact   with   it,  must,  §  217  and  Equations  (332),  be  normal  to   the  sur 
face.     That   is,  the  pressure  of  a  fluid  against  any   surface,  acts  always 
in    the   direction    of  the    normal. 


280 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


MOTION   OF   THE   FLUID   PARTICLES. 

§249. — The  particles  of  a  fluid  having  the  utmost  freedom  of 
motion  among  one  another,  all  the  forces  applied  at  each  particle 
must  be  in  equilibrio.  Regarding  the  general  Equation  (40)  as  ap 
plicable  to  a  single  particle,  whose  co-ordinates  are  x,  y,  z,  we  shall 
have 

and  supposing  the   particle   to    have    simply  a   motion  of  translation, 
we  also  have 

<$<p  =  0;   f^  =  °;   s*  =  °; 


and    that   equation   becomes 


( 


2  P  cos  a  —  ill' 


(Px 

d2y 


)  Sx 


+    (zPcosP-m-j^Sy    [   =  0  ; 
-\-    (2  Pcos  y  —  m-  ——^  J  o  z 
whence,  upon    the    principle  of  indeterminate   co-efficients. 


2  P  cos  a 


tPx 


2  P  cos  /3  —  w  •  -^  =  0 


2  P  cos  y  — 


;» 


d*2 
~dP 


=  0. 


(398) 


Now  the  terms  2  P  cos  a,  2  P  cos  /3  and  2  P  cos  7,  are  each  composed 
of  two  distinct  parts,  viz. :  1st.,  the  component  of  the  resultant  of 
the  forces  applied  directly  to  the  particle ;  and  2d.,  the  component 
of  the  pressure  transmitted  to  it  from  a  distance,  arising  from  the 
forces  impressed   upon   other   particles. 

Denote  by  X,   Y  and  Z,  the  accelerations,  in  the  directions  of  the 
axes   x,  y,  z,    respectively,    due  to   the  forces    applied   directly  to    the 


MECHANICS    OF    FLUIDb. 


281 


particle ;  then  w?,  being  the  mass   of  the  particle,  the  components  of 
the  forces  directly  impressed  will  be 

mX\     m  Y;     mZ. 

The  pressure  transmitted  will  depend  upon  the  particle's  place, 
and  will  be  a  function  of  its  co-ordinates  of  position.  Denote  by  p, 
the  pressure  upon  a  unit  of  surface,  on  the  supposition  that  every 
point  of  the  unit  sustains  a  pressure  equal  to  that  communicated  to 
the    particle  from  a   distance ;    then,  for  a  given  time,  will 

Conceive  each  particle  of  the  fluid  to  consist  of  a  small  rectan- 
gular parallelopipedon  whose 
faces  are  parallel  to  the  co- 
ordinate planes,  and  whose  con- 
tiguous edges  at  the  time  t, 
are  d  *,  dy  and  dz ;  and  let 
x,  y,  2,  be  the  co-ordinates  of 
the  molecule  in  the  solid  an- 
gle nearest  the  origin  of  co- 
ordinates. Then  would  the 
difference  of  pressure  on  the 
opposite  faces,  which  are  paral- 
lel  to  the   plane   zy,  were   these  faces   equal   to   unity,  be 

dp 


F (x  +  dx,  y,z,)  -  F (x,  y,  *,)  =—  •  dx; 

and   upon  the   actual    faces   whose   dimensions    are  each   dz.dy,  this 
difference   becomes,  Equation  (397), 

dp 


dx 


dx-dy  •  dz. 


In  like  manner  will  the  difference  of  the  pressures  transmitted 
to  the  opposite  faces  parallel  to  the  planes  zx  and  xy,  be,  respeo 
tively, 

d  y  '  dz  '  dx,     and     -£  *dz  •  d x  •dy. 


dy 


dz 


282 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


These  pressures  being  normal  to  the  surfaces  to  which  they  are 
respectively  applied,  they  will  act,  the  first  in  the  direction  of  *, 
tha  second  in  the  direction  of  y,  and  the  third  in  the  direction 
of  z.  And  as  these  differences  alone  determine  that  portion  of  the 
n  >tion    due  to    the  transmitted    pressures,  we    have 

_  ±        dp 

2  P  cos  a  =  wJl • —  •  dx  .d  y  .  dz  • 

dx 


2  P  cos  P  =miY  — 


dp 
dy 


dy . dx . dz  ; 


dp 


2  P  cos  y  =  m  Z — —  -  dz  .  dx  .  dy. 

Danote  by  D  the  density  of  the  mass  ;w,  then  will,  Equation  (2), 

m  —  D  .  dx  .dy  .  dz, 

and    by  substitution,  E< fixations  (398)  become 


1 

ii 

D 

dx 

1 

■     ■ 

D 

dp 
dy 

1 
D 

dp 
dz 

dp' 

=  "r  -  ^i W 

dp  ' 

_    7        d*z 

-  L  ~  JW  ; 


(399) 


Denote  by  «,  v  and  w,  the  velocities  of  the  molecule  whose  co- 
ordinates are  xyz,  parallel  to  the  axes  x,  y,  z,  respectively,  at  the 
time  t.  Each  of  these  will  be  a  function  of  the  time  and  the  co- 
ordinates of  the  molecule's  place;  and,  reciprocally,  each  coordinate 
wil    be  a  function  of  t,  u,  v  and  w\  whence,  Equations  (12)  and  (13), 


d2  x        d  u 


i2  x        du        /du\     dt       du    dx       du     dy       du     dz 
as as   I  —  ) 1 .  -Z  J .  —  • 

dt2        dt         \dt/     dt       dx    dt        dy      dt       dz      dty 


dx     dy     dz 
and  replacing  — ->>    — -?   —  >    by  their  values  u,  v,  w.  respectively,  w« 

(XL        CL  Z       %M>  £ 


have 


(Px 


x         /du\         du  •  du  du 


MECHANICS     OF    FLUIDS. 


283 


in    the   same  way, 
d 


dv 
.  Ht 

dz        ' 


1  if  /  dv\         d  v  dv 

7T  =  (-77)  +  I-  '  »  +  T-  *  v  + 

^2  \  dl  /         dx  dy 

d2  z  /  dw\         dw  dw  dw 

_^_    3^  I    I  -j-    .    1/     -J-     .    |»     -1_      • 

d(l  \  d  t  /  dx  dy  dz 


w 


which,  substituted  in    Equations  (399),  give 


1 

u 

dp 
dx 

1 

a 

dp 

dy 

1 

D  ' 

dp 
d  z 

X 
Y 
Z 


/  du\ 
\dt) 

(—) 

\dt  / 

/dio\ 
\dl) 


d  u 
dx 

dv 
dx 

dw 
dx 


u 


u  — 


u  — 


d  u 

dy 

V 

dv 

dy 

V 

dw 

dy 

V  ■ 

five 

un 

du 
~dz~ 

dv 
dz 

dw 
dz 


w 


; 


w ; 


w. 


(400) 


Here  are    three  equations  involving  five    unknown  quantities,  viz. 
«,  v,  ic,  p  and  D,  which    are   to   be  found  in    terms  of  x,  y,  z  and  /. 

Two  other  equations  may  be  found  from  these  considerations,  viz  : 
the  velocity  in  the  direction  of  x,  of  the  molecule  whose  co-ordinates 
are.  x  y  z,  is  v ;  the  velocity  of  the  molecule  in  the  angle  of  the 
parallelopipedon  at  the  opposite  end  of  the  side  dx,  at  the  same  time, 

is 

du 

u  H — —'dx: 

dx 

and    hence   the   relative    velocity    of  the   two  molecules   is 


du     .  du     . 

u  -\-  - —  dx  —  u  =  - —  -d x. 
dx  dx 


At    the   time   t,  the   length   of    the   edge    joining  these    molecules    ** 
dx,  and   at   the   end   of  the   time   t  +  d  t,  this  length  will    be 

1  du      .        .  ,/,        du 

d  x  -|-  -, —  •  dx  .  d  t  =  d  x  (  1  -f-  -r—  '  at): 
dx  v  dx 

the    second   term    being    the    distance    by    which    the    molecules    in 

question    approach    toward    or     recede     from     one     another    in    the 

lime  dt. 

18 


284:    ELEMENTS  OF  ANALYTICAL  MECHANICS. 

In  the  same  way  the  edges    of  the   parallelopipedon  which  at    the 
fcime  t,  were  dy  and  dz,  become  respectively, 

,  d  v      .        .  ,     /,         dv 

dW  7  7  7  /I  .  ^W  7      N 

d  z  -\ — - —  •  d  z  .  d  t     =  dz  [l  -j — - —  •  dt)\ 
d  z  d  z 

and   the    volume   of  the  parallelopipedon,  which    at   the   time   t,    waa 
dx  .dy  .dz,  becomes  at   the    time  t  -f-  dt, 

dx.d„.dz(x+^.d().(1  +  ±L.dt).(1+i£.dt). 

The   density,   which   was   D,  at   the    time  t,  being  a  function  of  xyz 
and  t,  becomes   at   the   time   t  -\-  dt, 

dD     .         dD     ,  d.D     ,         dD     . 

D  +  -r-r'dt  +  -J—  -dx  +-T—  -dy  -f  ——  *dz\ 
dt  dx  dy  dz 

which   may   be    put   under   the   form, 

{dD         dD   dx         dD    dy         dD    dz\   , 

D  4-  1 1 4-  •  -—  4- I  d  t ; 

^  Vtf*    T    dx     dt   ^    dy     dt    T    dz     rf#/       ' 

and  replacing 

dx        dy        dz 
dt         dt         dt 

by  their  values  u,  v,  w,  respectively, 

'rfi>         rfi)  dD         .    dD 

y 

Multiplying  this  by  the  volume  above,  we  have  for  the  mass  of    th* 
parallelopipedon,  which  was 

D  .dx  .dy  .dz, 

at  the   time    t,  the  value, 


/dD         dD  dD  dD      \    , 

D  4-  (-77     +  -7—  •«  +  -j—  -v  -f  -j--w)  dt. 
\  dt  dx  dy  dz       J 


r  (dD       dD  dD  dD      \   ,  1 


X  dx.dy.dz  (l  4-  -J"d<)  *  (]  +  ~d~'dt>)  '  0  +  ~T~'dt) 
at   the  time    *  -}-  rf  & 


MECHANICS    OF    FLUIDS.  285 

But  these  masses  must  be  equal,  since  the  quantity  of  matter 
is  unchanged.  Equating  them,  striking  out  the  common  factors,  per- 
forming the  multiplication,  and  neglecting  the  second  powers  of  the 
differentials,  we  have 

„  /dn         dv        dw\       dD       dD  dD  dD 

D  (__  + -—  +  -_)  +_.+  _.«+_  .»+  —  »==  0.(401) 

\dx  dy         dz  '  dt         dx  dy  dz 

This  is  called  the  Equation  of  continuity  of  the  fluid.  It  expres 
ses  the  relation  between  the  velocity  of  the  molecules  and  the  den 
sity  of  the  fluid,  which  are  necessarily  dependent  upon  each  other. 
This  is  a  fourth  equation. 

§250.— If  the  fluid  be  compressible,  then  will  the  fifth  equation 
be   given   by  the   relation, 

F{D,p)  =0,  •    •    • (402) 

as  is  illustrated  in  the  particular  instance  of  Mariotte's  law,  Equa- 
tion (389).  The  form  of  the  function  designated  by  the  letter  F 
will    depend    upon    the   nature   of  the    fluid. 

§251. — If  the  fluid  be  incompressible,  the  total  differential  of  D 
will   be   zero,  and 

dD         dD  dD  dD 

a  t  dx  ay  dz  s 

and  consequently,  the  equation  of  continuity,  Equation  (401),  becomes, 

d  u  dv  dw  ^        v 

-I7+7j  +  -J7=0> <40*> 

aid  we  have  for  the  determination  of  ?/,  t»,  w,  D  and  />,  the  five 
Equations  (400),  (403),  (404). 

§  252. — These  equations  admit  of  great  simplification  in  the  case 
of  ai.  incompressible  homogeneous  fluid  when  u-dx  -f-  v.dy  -f-  w.dz, 
is  a  perfect  differential.     For  if  we  make 

ud  x  -f-  vdy  +  wdz  =  d<p. 


'286  ELEMENTS    OF    ANALYTICAL    MECHANICS, 

then    from  the   partial  differentials  will 

d  cp  d  cp  d  qp  ,«„» 

w  =  — —  ;     v  —  — — ;     w  =  — — ;    ....     (40o) 

dx  dy  dz  N       ' 

*hich,  in  Equation  (404),  gives  for   the  equation   of  continuity, 

d^_        d*?        d?v  

dx2    ^  dy2  ^    dz2  '  {       } 

by    the   integration  of  which  the   function  cp  may    be   found. 
Differentiating  the  values  of  u,  v  and  w  above,  we  have 

.  d2  cp  cPcp  d2cp 

du  =  — —  :    o  =  — = —  ;    d  w  =  — —  • 
dx  dy  dz 

Eliminating  u,  v,  w,  d  u,  d  v  and  dw,  from  Equation  (400),  by  means 
of  the    values  of  these  quantities  above,  we  have 

\      d  p  d2cp  d  cp    d2cp         dcp       d2  cp  d  cp        d2  cp 


D 

dx 

1 

u 

dp 
dy 

1 

dp 

=  Y  - 


=:    Z 


dx'dt  dx  d  x2        dy     dx.dy  dz     dx.dz* 

d2 cp  dcp       d2 cp  d<p    d2  cp  dcp        d2cp 

dy.dt  dx  dy  .dx       d  y   d  y2  dz     dy.dz^ 

d2  cp  dcp       d2  cp  d  cp       d2  cp  dcp    d2  cp 


D      dz  dz.dt        dx    dz.dx        dy    dz.dy        dz     dz2 

Multiplying  the  first  by  dx,  the    second    by  dy,  the  third  by  dz, and 
aiding  we   mid, 

.■  •■    - 

% 

l  Ficrj  which,  by  integration,  may  be  found  the  pressure  at  any  point 
of  an  incompressible  fluid  mass  in  motion,  when  Equation  (406)  is 
the    equation  of  continuity. 

§  253. — When    the    excursions    of    the    molecules    are    small,    the 
second    powers  of  the  velocities  may   be  neglected,  which  will  reduce 
■    Equation  (407)  to 

* 

\j-dp  =  Xdx  +  Ydy  +  Zdz  -   d  ^| .     .     .     (408) 


MECHANICS     OF    FLUIDS.  287 

§254. — If  the  condition  expressed  by  Equation  (406)  be  not  ful- 
filled, then  we  must  have  recourse  to  Equation  (404)  to  find  the 
pressure. 

§255. — Resuming  Equation  (401),  which  appertains  to  a  compre* 
sible  fluid,  retaining   the    condition  that  .    . 

udx  +  v  dy  +  wdz  =  dtp 
is   a   perfect  differential,  and  from  which,  therefore, 

dx   '  dy  '  dz   '  v       v 

we    obtain  by  substitution, 

■    (   du  dv        dw  )        dD      dD  dip        dD  dtp       dD  dtp 

(    dx  dy        dz    )         dt        dx    dx         dy    ay        dz    dz 

If  the    excursions  of  the   molecules   from    their   places   of  rest  be 

very    small,    both    the    change    of    density    and    velocity    will    be  so 

small    that   the    products   which    constitute    the   last   three   terms  of 

this    equation    may   be  neglected,  and   the  t  equation   of  continuity  be 
comes 

_     / du  dv  dw\        d  D 

D  '  \—j — Y  i — Y  -i~)  +  ~jt  =  ° » 

V  dx  dy  dz  /  dt 

and  replacing  du,  d  v  and  d  w,  by  their  values  from  Equations  (409), 
and  dividing  by  D,  we  find 

Ug*  +  **  +  f*  +  .£*  =  o.  •  •  •    (4ro) 

d  t  dx2         dyz  dz1 

■ 

from  which,  and  Eq.  (408),  the  equation  connecting  the  extraneous 
forces  with  the  co-ordinates  xyz,  and  that  expressive  of  Mariotte's 
law,  the  function  9  may  be  found,  then  the  value  of  2),  and  finally 
that  of  p.  ;, 

The  excursions  being  small,  if  we  impose  the  additional  condi- 
tion  that   the   molecules  of   the   fluid   are   not   acted    upon    bv    extra- 


288         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

neocs  forces,  in  which   case   the   motions   can   only  arise   from    some 
arbitrary   initial  disturbance;    then,  Equation  (408), 

D       r  dt  dt 

and  by  Mariotte's  law, 

p  =  P.D  =  a2.D (411) 

from  which 

dp  =  a?d£> (412) 

and  the  above  may  be  written,  after  dividing  by  d  t, 

t_td_D  d\ogD  d^ 

dt     D  {dt  dt'  v      ' 

which,  in  Equation  (410),  gives 

^1  =  a*  (dI±  +  **  +  ^L)   .     .     .     .     (414) 
dt2  \dx3    T  dy*   T   dz*/  \       ' 

'}  From  this  Equation  the  function  <p  is  to  be  determined,  then  the 
value  of  7),  from  Equation  (410),  and  that  of  p,  from  either  of  the 
Equations  (411)  or  (413).  ; 

§256. — Conceive  a  homogeneous  elastic  fluid  to  be  disturbed  at  one 
of  its  points  by  the  sudden  expansion  or  contraction  of  the  element 
there  situated.  This  will  break  up  the  equilibrium  of  the  surrounding 
molecular  forces  at  that  point,  the  particles  adjacent  will  move  to  re- 
store the  uniformity  of  density,  and  an  expanding  disturbance  will  pro- 
ceed outward  from  this  as  a  centre.  Take  the  origin  at  the  point,  and 
denote  the  distance  of  any  particle  involved  in  the  disturbance,  at  any 
time  t,  subsequent  to  the  disturbance  by  r,  then  will 


x1  +  y1  +  z1  =  r\ 

Denote  the  velocity  of  the  particle,  supposed  in  the  direction  of  r,  bj 
f;  then  will 

«  =  £.-:     «  =  £.-;     ?/>  =  £.-. 

/♦  r  r 


MECHANICS    OF    FLUIDS.  289 

Differentiating  the  first  of  the  above  equations,  we  have 

xdx-\-ydy-\-zdz  —  r  .  dr. 

Substituting  the  values  of  a-,  y,  and  z  from  the  second,  third,  and  fourth, 
there  will  result 

udx  -\-  vd y  +  w dz  =  C,  .  dr\ 

go  that  this  satisfies  the  condition  of  the   first   member  beinjr  an  exact 
differential ;  and,  therefore,  d  $  =  £.  d r ;  or 

*""  dr' 

And  hence 

d  cp       d  «p     x  d  (p       d  <p    y  d  $       d  q>     z 

d  x       dr      r '  t/  y        dr    >r'  dz       dr     r ' 


differentiating, 


o?2  <p       d'2  (p    x*       d  $    y'2  -4-  2* 


rf8  <p  G?2  (p  J/2  C?  (p  Z2  -J-  X9 

d  y*  d  r*  r1  c/  r  r3      ' 

rf2  <p  a?2  <p  z2  (/  (p  a:2  -|-  y* 

d  z2  (/  r1  r1  d  r  r 

and  these  values,  substituted  in  Equation  (414),  give 

d2  (p  q  (d?  <p  t    2  d  <p ' 

which   may  be  written, 


?  -  a2  (—       2     — \  * 
8  \  d  r8       r     d  rj  ' 


d*  r  <p  „    rf8  r  (p  ,- 

-ir  =  *—j? <414>' 

of  which  the  integral  is,   Appendix  No.   IVn 

r?  =  ^r  +  a/)+/(r-fl<); 

and  in  which   ^  and  /  denote  any  arbitrary  functions  whatever.     From 
this  we  have 


290  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

9  =  -lF(r  +  at)+f(r-at)]     ••..     (415) 

Taking  the  first  differential  coefficient  of  <p  with  respect  to  r,  and  re- 
placing its  value  by  £, 

^l.^Jf  +  a  t)  +/'(r  -at)]-  1  [F(r  +  a  t)  +  f(r  -  a  t)]. 

For  any  considerable  distance  from  the  origin,  the  second  term  may  be 
omitted  in  comparison  with  the  first ;  and  there  will  result,  alter  squaring 
ani  multiplying  by  m,  the  mass  of  the  moving  particle, 

7)1 

m?  =  -.[F!(r  +  at)+f(r-at)y     .     .     .     (416) 

The  first  member  is  the  living  force  of  the  moving  particle,  or  double 
the  quantity  of  work  it  may  impress  upon  the  organs  of  sense  exposed 
to  its  action.  The  effect  it  may  produce  will,  therefore,  all  other  things 
being  equal,  vary  inversely  as  the  square  of  its  distance  from  the  place 
of  primitive  disturbance.  Equations  (415)  and  (416)  are  employed  in 
discussing  the  theory  of  sound. 


EQUILIBRIUM    OF    FLUIDS. 


§  257.— If  the   fluid  be   at   rest,  then  will 

d2x        _      d2y        '      dPz 


d  t2  ~~  "  '    d  t2 
and   Equations  (399)  become 

dp 


dfl 


d  x 

dp 
dy 

dp 
dz 


=  D.X; 

mA.Yy 

=  D.Z. 


(417) 


§258. — Multiplying  the  first   by  dx,  the  second    by  d  y,  the  third 
by   dz,  and  adding  we  find, 

dp  =  D  (Xdx  +  Ydy  -I-  Zdz); 


•     •     •     • 


(418) 


MECHANICS    OF    FLUIDS.  291 

and    by  integration, 

p  =  JD  .{Xdx  -f  Ydy  -f-  Zdz)-,    .     .     .     .(410) 

whence,  in  order  that  the  value  of  p  may  be  possible  for  any 
point  of  the  fluid  mass,  the  product  of  the  density  by  the  function 
X d x  -f-  Ydy  +  Zdz,  must  be  an  exact  differential  of  a  function  of 
the  three  independent  variables  x,y,z.  Reciprocally,  when  this  condi- 
tion is  fulfilled,  not  only  will  the  pressure  at  any  point  become  known 
by  substituting  its  co-ordinates,  but  the  Equations  (417),  will  be  sat- 
isfied, and  the  fluid  will  be  in  equilibrio. 

§  259 — Conceiving  those  points  of  the  fluid  which  experience  equal 
pressures  to  be  connected  by,  indeed  to  form  a  surface,  then  in 
passing  from  one  point  to  another  of  this  surface,  we  shall  have 
dp  ss  0,  and 

Xdx  +  Ydy  +  Zdz  =-'0; (420.) 

which  is  obviously  the  differential  equation  of  an  ,cquipotcntial  or  level 
surface. 

Dividing  this  by  R  d  s,  in  which  mR,  denotes  the  resultant  of  the 
forces  which  act  upon  any  particle,  and  ds,  the  element  of  any 
curve  upon  the   surface  passing  through    the   particle,  we  havp 

Xdx  Y   dy         Zdz 

-TT-T-  +  -$—  +  7J--T-  =0;  .     .     .     .(421 
R    ds  R    ds  R     ds  v 

whence  the  resultant  of  the  forces  acting  upon  any  one  of  the 
elements  of  a  surface  of  equal  pressure,  is  normal  to  that  surface. 
This  is  the  characteristic  of  what  is  called  a  level  surface,  which 
may  be  defined  to  be  any  surface  which  cuts  at  right  angles  the 
direction   of    the  resultant  of  the  forces   which  act  upon  its  particles. 

§260. — If  Equation  (420)  be   integrated,  we   have 

f(Xdx  +   Ydy  +  Zdz)  =  C.       ....  (422) 

in  which  the  potential  C  is  the  constant  of  integration.  The  magni- 
tudes of  this  constant  must  result  from  the  dimensions  of  the  surface, 
or  from  the   volume    of    the    fluid    it    envelops..    By  giving  it  different 


292         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and  suitable  values,  we  may  start  from  a  single  particle  and  proceed 
outwards  to  the  boundary  of  the  fluid,  and  if  the  successive  values 
differ  by  a  small  quantity,  we  shall  have  a  scries  of  level  concentric 
strata. 

The  last  value  for  the  potential  C  determines  the  exterior  or 
bounding  surface  of  the  fluid ;  because  this  surface  being  free,  the 
pressure  upon  it  will  be  zero ;  the  differential  of  the  pressure  from  one 
point  to  another  will,  therefore,  be  zero,  and  the  differential  equation 
will  be  that  numbered  (420),  or  that  of  equal  pressure.  Every  free  sur- 
face of  a  fluid  in  equilibrio  is,  therefore,  a  level  or  equipotential  surface. 

§261.— Putting  Equation  (418)    under   the  form 

-^  =  Xdx  +  Ydy  -f-  Zdz, (423) 

we  see  that  whenever  the  second  member  is  an  exact  differential. 
p  must  be  a  function  of  D,  since  the  first  member  must  also  be  an 
exact   differential.     Making,  therefore, 

p  =  F(£>), (424) 

in  which  F  denotes  any  function  whatever,  the  above  equation  be- 
comes 

—j^  =  Xdx  f  Ydy  +  Zdz-,      .     .     .     (425) 

but  for  a  level  surface  or  stratum,  the  second  member  reduces  to 
tero ;  whence, 

dF(D)  =  0-, 
and    by   integration, 

F(D)  =  0; 

whence,  not  only  will  each  level  stratum  be  subjected  to  an  equal 
pressure  over  its  entire  surface,  but  it  will  also  have  the  same 
density  throughout. 

§262. — If  the  fluid  be  homogeneous  and  of  the  same  temperature 
throughout,  then  will  D  be  constant,  and  the  condition  of  equilibrium 


MECHANICS    OF    FLUIDS.  293 

simply  requires  that  the  potential  function  Xdx  -f  Ydy  +  Z  dz,  Equa- 
tion (419),  shall  be  an  exact  differential  of  the  three  independent 
variables  #,  y,  z,  and  when  this  is  not  the  case,  the  equilibrium 
will  be  impossible,  no  matter  what  the  shape  of  the  fluid  mass, 
and    though  it   Mere   contained   in   a   closed    vessel. 

But  the  function  above  referred  to  is,  §  133,  always  an  exact 
differential  for  the  forces  of  nature,  which  are  either  attractions  or 
repulsions,  whose  intensities  are  functions  of  the  distances  from  the 
centres  through  which  they  are  exerted.  And  to  insure  the  equi- 
librium, it  will  only  be  necessary  to  give  the  exterior  surface  such 
shape  as  to  cut  perpendicularly  the  resultants  of  the  forces  which  act 
upon  the  surface  particles.  This  is  illustrated  in  the  simple  example 
of  a  tumbler  of  water,  or,  on  a  larger  scale,  by  ponds  and  lakes 
which  only  come  to  rest  when  their  upper  surfaces  are  normal  to 
the  resultant  of  the  force  of  gravity  and  the  centrifugal  force  arising 
from    the    earth's  rotation    on    its  axis. 

In  the  case  of  a  heterogeneous  fluid  subjected  to  the  action  of  a 
central  force,  its  equilibrium  requires  that  it  be  arranged  in  concentric 
level  strata,  each  stratum  having  the  same  density  throughout.  And 
the  equilibrium  will  be  stable  when  the  centre  of  gravity  of  the 
whole  is  the  lowest  possible,  §  136,  and  hence  the  denser  strata  should 
be   the   lowest. 

When  thf>  fluid  is  incompressible,  the  density  may  be  any  function 
whatever  of  the  co-ordinates  of  place.  It  may  be  continuous  or  dis- 
continuous. When  it  is  given,  the  value  of  the  pressure  is  found  from 
Equation  (419). 

§  2G3. — In    compressible   fluids   the   density    and  pressure  are    con- 
nected  by    law,  and    the   former  is  no  longer  arbitrary. 
Dividing  Equation  (418)  by  Equation  (389),  we  have 

dp_  _  Xdx  +  Ydy  +  Zdz       #        #     §      /425)' 
p     ~  P 


Integrating, 


PXdx  +   Ydy  +  Zdz        .       „  ,A- 


294         ELEMENTS     OF    ANALYTICAL    MECHANICS, 
denoting  the  base  of  the  Naperian  system  by  e,  we  have 

fXdx  +  Ydy+Zdi  .  (  \?><y\ 

P=c.eJ — XJrm — I f**J 

and    this  substituted  in  Equation  (389).  gives 


fXdx+  Ydy+  Zdz 

Ce               P 
D  =  — a (428) 


These  equations  determine  the  pressure  and  density. 

For  any  surface  of  constant  pressure,  the  exponent  of  e,  in  Equa- 
tion (427),  must  be  constant,  its  differential  must,  therefore,  be  zero, 
and  all  the  consequences  deduced  from  Equation  (420)  will  follow  ; 
that  is,  when  the  fluid  is  at  rest,  it  must  be  arranged  in  level  strata, 
each  stratum  having  the  same  density  throughout,  with  the  addition 
that  the  law  of  the  varying  density  must  be  continuous  by  the  re- 
quirements of  Mariotte's  law. 

If  the  temperature  vary,  then  will  P  vary,  and  in  order  that 
Equation  (425)'  may  be  an  exact  differential,  P  must  be  a  function 
of  x  y  z,  and  hence,  Equations  (427)  and  (428),  when  p  is  constant, 
D  will  be  constant;  that  is,  each  level  stratum  must  be  of  uniform 
temperature  throughout. 

It  is  obvious  that  the  atmosphere  can  never  be  in  equilibrio  ;  for 
the 'sun  heating  unequally  its  different  portions  as  the  earth  turns 
upon  its  axis,  the  layers  of  equal  pressure,  density  and  temperature 
can  never  coincide.  Hence,  those  perpetual  currents  of  air  known  as 
the  trade  winds,  and  the  periodical  monsoons ;  also,  the  sea  and  land 
breezes,  variable  winds,  &c,    &c. 

§  264. — Rest  is  a  relative  term ;  when  applied  to  a  particle  of  a 
fluid  mass,  it  means  that  that  particle  preserves  unaltered  its  place  in 
regard  to  the  other  particles;  a  condition  consistent  with  a  bodily 
movement  of  the  entire  mass. 

If  a  liquid  mass  turn  uniformly  about  an  axis,  the  preceding 
equations  will  make  known  its  permanent  figure.  For  this  purpose 
it  will  be  sufficient  to  join  to  the  forces  X,  F,  Z,  the  centrifugal  force 


MECHANICS     OF    FLUIDS. 


2% 


Take    the  axis  z  as  the  axis  of  rotation ;    denote  the  angular  velocity 
by  <p,  and    the  distance  of 
the    particle    M   from    the 
axis  z  by  r;  then  will 

?.2    _    x2    _J_    y2  . 

tne  centrifugal  force  of  M 
regarded  as  a  unit  of  mass, 
will  be 

and  its  components  in  the 
direction  of  x  and  y,  respectively, 

2     x 

r  .  en*  •  —  =  zcp  ; 

r 

2  y        2 

r 

and  these  in    Equation  (418),  give 

dp  =  D.(Xdx  +  Ftfy  +  Zrf*  -r  q>2.xdx  +  <p2y  .</#)••  (420) 

When  the  second  member  is  an  exact  differential,  the  permanent  form 
will   be   possible. 

For   the    free    surface  dp  =  0,  and  we  have 

Xdx  -h   Fo?2/  +  Zdz  +  <p2.*.tfz  -f  <p2yc?y  =  C  •  -(430) 

Example  1. — Let  it  be  required  to  find  the  figure  assumed  by 
the  free  surface  of  a  heavy  and  homogeneous  fluid  contained  in  an 
open  vessel  and  rotating  about   a  vertical    axis. 

Here, 

X  =  0;      V=0;     Z=  -g- 

and    Equation  (430)  becomes 

gdz  =  to2  (xdx  -f  ydy). 
Integrating, 


r 


*  =  ^  (*»  +  *)  +  c> 


(431) 


which  is  the  equation  of  a  paraboloid  whose  axis  is  that  of  rotation. 


296 


ELEMENTS     OF    ANALYTICAL     MECHANICS. 


To  find  the  constant  6',  let  the  vessel  be  a  right  cylinder,  with 
circular  base,  whose  radius  is  a,  and  denote  by  h  the  height  due  to 
the    velocity  of  the  fluid  at  the  circumference,  then 


and 


«2<p2  =  2gh, 


h.r2 
a1 


(432) 


Denote  by  6  the  height  of  the  liquid  before  the  rotation ;  its 
volume  will  be  ira2.b.  Conceive 
the  whole  body  of  the  liquid  to 
be  divided  into  concentric  cylin- 
drical layers,  having  for  a  common 
axis  the  axis  of  rotation.  The  base 
of  any  one  of  these  layers  will 
have  for  its  area,  neglecting  dr2, 
2irr.dr,  and  for  its  volume,  taking 
the  origin  of  co-ordinates  in  the 
bottom  of  the  vessel,  2irr .dr.  z, 
which  being  integrated  between  the 
limits  r  =  0  and  r  =  at  will  give 
the  whole  volume  of  the  fluid,  and 
hence, 


a2b 


=  2f\r.dr; 


replacing  r .  d r  by  its  value  from  Equation  (432),  and  integrating 
between  the  limits  z  =  C  and  z  =  h  -f-  C,  which  are  the  values 
given   by  Equation  (432)  for  r  =  0   and  r  =  a,  we  find 

C  -  b  -  ±h, 

and    the   equation  of  the   upper   surface  becomes 


z  = 


a< 


+  b  -  \h. 


The   least   and  greatest   values   for   z,  are    b  —  \h    and    b  -f  \  K 
obtained  by  making  r  =  0  and  r  =  a,  so   that  the  depression  nl  the 


MECHANICS    OF    FLUIDS. 


297 


liquid  at  the  axis  is  equal  to  its  elevation  at  the  surface  of  the 
cylindrical  vessel,  and  is  equal  to  half  Yhe  height  due  to  the 
velocity  of  the  latter. 

§  265. — Example  2. — Let 
the  fluid  elements  be  attract- 
ed to  the  centre  of  the  mass 
by  a  force  varying  inversely 
as  the  square  of  the  distance. 
Take  the  origin  at  the  cen- 
tre ;  denote  the  distance  to 
the  particle  m  from  that  point 
by  r,  and  the  intensity  of  the 
attractive  force  at  the  unit's 
distance  by  k.     Then  will 


x 


y 


P  =  m  — ;     cos  a  —  —  —  ;     cos  p  ss ;     cos  y  =  — 


z 


r2>     —  r  »     ~-  r  >     — i  -  rf 


and 


r3 


kg 

r3  ' 


which  in  Equation  (430),  give 

k 

—  {xdx  +  ydy  +  zdz)  —  <f(xdx  +  ydy)  -  0, 


or 


*£_£<,<*  4,^  =  0, 


and  by  integration, 


-  +  V  (*2  +  y»)  =  C; 


making 

x2  -f  y2  =  r2cos24, 

« 

in  which  5  denotes   the  angle  made  by  r,  with   the  plane  zy, 

A  +  ?_.  ^cos'd  =  r, 

r  2 


298  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and    denoting    the    distance    from    the    origin    to    the    point   in    which 
the  free  surface  cuts  the  axis  z  by  unity,  we  have,  by  making  &  ==  90°, 

1 

vhich    substituted   above,  and    solving    with  respect   to  cos2d,  gives 

i<p2-cos»4  =  k(r  -1)  ,     ......     (434) 

and    making   r  =  1  -\-  u,  we  have 

ku 


\  p2  •  cos2  6 


(1  +  uf 

if  the  angular  velocity  be  small,  then  will  u  be  very  small. 
Developing  the  second  member  with  this  supposition,  and  limiting 
the    terms   to    the  first   power  of  »,  we  find 

]  p2  •  cos2  6  =  k  (u  —  3  w2).  •     •     •   '  •     •     (434)' 

Neglecting  3  w^,  and  replacing  w  by  its  value,  viz.:  r  —  1,  we 
have    for   a    first   approximation, 

<P2 
rrl  +  ~.  cos2  d. 
2  a; 

From  Equation  (434)',  we  find 

a>2  •  cos2  &        L    . 

and   this   in   the    equation 

rsl+u, 
gives  .    •  . 

r  r=  1  +  -?- .  cos2  H3m2; 
2k 

and  replacing  w2  by  its  approximate  value   -  — >i    above,  by  neg- 

lecting 3  m2,  we  have 

,    ,    92         o *    ,    3  <p4  •  cos4  4 

for   the   polar  equation  of  the   meridian  section. 


MECHANICS    OF  FLUIDS.  299 


Comparing  this  with  the  equation 
1 


r  = 


_=  =  1  +  £t2cos20  +  $.e*.cos40  4-  &c, 
\/l  —  e2  cos2  0 

they  become  identical  by  neglecting  the  higher  powers  and  making 


*' 


The  free  surface  of  the  fluid  approximates,  therefore,  very  closely  to 
an  ellipsoid  of  revolution  of  which  the  eccentricity  of  its  meridian  section 
is  equal  to  the  square  root  of  the  quotient  arising  from  dividing  the 
centrifugal  force  at  the  unit's  distance  from  the  axis  of  rotation,  by  the 
force  of  attraction  at  an  equal  distance  from  the  centre. 


PRESSURE     OF     HEAVY     FLUIDS. 

§  266. — When  a  fluid  is  acted  upon  by  its  own  weight,  if  the  axis  z 
be  taken  vertical  and  positive  upwards,  then  will 

and  the  equation  of  a  level  surface,  Equation  (420),  becomes 

—  gdz  =z  0  ; 

or  the  potential  gz,  and  consequently  z  itself,  is  constant.  All  such  sur- 
faces arc,  therefore,  horizontal  and  parallel ;  which  result  may  be  clearly 
seen  from  the  fact  that  they  are  cut  at  right  angles  by  the  force  of 
gravity. 

Also,  the  general  equation  (418)  becomes 

dp  +  Dcjdz  —  0, (435) 

which  is  true  both  for  liquids  and  gases. 

For  homogeneous  liquids,  the  specific  gravity  Dg  may  be  supposed 
constant ;  denoting  it  by  cj  and  integrating, 

j)-f-(.)z  =  c; (435') 

which  may  be  written  thus : 

z  +  l  =  H, (435") 

6) 

a  constant  depth,  which  is  technically  called  the  total,  or  full,  dynamic 
head. 


300      ELEMENTS    OF    ANALYTICAL    MECHANICS. 


A 

n 

Ar 

r 

\n' 

B 

m 

\  77 

B' 

ySr 

DC 


0 


0' 


oc 


.   To  interpret  these  results,  let  the  co-ordinate  plane  of  xy  be  taken  at 
the  bottom  of  the  liquid ; 
and  let  m  be  the  position 
of  a  particle  at  any  height 
z  equal  to  Om ;  also,  let 

mn  equal  —  be  the  head, 

or  depth,  due  to  the 
pressure  p ;  then  Equa- 
tion (435")  shows  that 
the  sum  of  these  two 
heights  is  constant  and 
equal  to  the  full  head  H. 

Also,  Equation  (435)  may  be  constructed  by  drawing  the  oblique 
line  nc,  whose  abscissas  represent  the  pressures  corresponding  to  the 
heights,  or  ordinates  z.  This  line  cuts  the  axis  Oz  at  ft,  where  the 
pressure  is  zero ;  and  the  axis  Ox  at  c,  where  Oc  is  the  pressure  at  the 
bottom.  Its  direction  is  given  by  (*>,  the  specific  gravity,  which  is  also 
the  tangent  of  the  angle  en  0. 

That  the  term  —  represents  a  line  may  readily  be  seen,  for  p,  the 
(1) 

pressure  upon  the  unit  of  surface,  is  a  weight  w  divided  by  an  area  n'P ; 
and  the  specific  gravity  u  is  another  weight  w"  divided  by  a  volume  n"P. 
So  that 


p        hi'    to      P\  1  . 

.0)  \n       w       Pf 


in  which  the  coefficient  »,  within  brackets,  is  only  a  product  of  abstract 
numbers,  but  /  is  linear. 

Equation  (435'),  or  its  construction,  shows  the  law  that  in  heavy 
liquids  pressiwes  are  pi'oportional  to  depths. 

No  portion  of  the  liquid  can  be  above  the  level  AA\  where  z  is  equal 
to  H  and  p  is  zero ;  for  if  z  be  supposed  to  exceed  H,  then  either  p  or 

l 

(j  must  become  negative,  which  is  impossible. 

If  the  atmosphere  rests  upon  the  liquid,  then  its  surface  sustains  the 
pressure  p0  due  to  the  weight  of  the  air;  and  drawing  BB'  below  AA 
at  the  distance 

*>,.  —  P° . 
nr  =  —  ; 


MECHANICS    OF    FLUIDS.  301 

this  plane  BB'  will  be  the  top  of  the  liquid ;  above  wjiich  none  of  it 
can  exist  except  in  the  form  of  vapor  mixed  with  the  air. 

The  atmospheric  pressure  may  be  eliminated  by  transferring  the 
origin  of  co-ordinates  and  axis  of  z  to  the  new  position  O'n',  for  which  p 
is  zero  at  n.  And  generally,  whenever  in  hydraulic  questions  only  differ- 
ences of  liquid  pressure  need  to  be  considered,  the  atmospheric  -pressure 
may  be  eliminated. 

For  gases,  the  value  of  D  given  by  the  law  of  Gay  Lussac,  Equation 
(390),  may  be  substituted  in  Formula  (435) ;  and  thus,  after  dividing* 
we  find 

9  P 

If  in  this  0  be  constant,  we  may  write  A  for  the  factor  of  the  second 
term,  and  integration  gives 

z  -f-  A  \ogp  =  c ; 

for  any  other  height  z9  and  pressure  pg,  we  have, 

ze  +  A  \ogp0  =  c; 
and,  therefore, 

z  —  z„  =  A  log  — ; (436) 

an  equation  which  gives  the  altitude  z  —  z^  when  p,  p0,  and  A  have 
been  determined  by  observation. 

§  267. — Let  now  the  liquid,  acted  upon  by  its  weight  only,  be  con- 
tained in  any  vessel ;    and  let  the  axis  z  be  taken 
vertical  and  positive  downwards,  then 

X=0,         F=0;         Z=f/; 
and  Equation  (418)  becomes,  after  integrating, 

p  =  £ffz+  C\ 
and  assuming  the  plane  xy  to    coincide   with    the 
upper  surface  of  the   fluid,  which    must,  when    in 
equilibrio,  be  horizontal,  we  have,  by  making   Z  =  0, 

p'  =  C; 

in  which  p'  denotes  the  pressure  exerted  upon  the  unit  of  the  free  Furfacc. 

Whence, 

p  —  p'=zD.n.z (437) 


"302       ELEMENTS    OF    ANALYTICAL    MECHANICS 


The  first  member  is  the  pressure  exerted  upon  a  unit  of  surface,  every 
point  of  which  unit  sustains  a  pressure  equal  to  that  upon  the  element 
whose  co-ordinate  is  z. 

If  p'  =  0    k  the  pressure  on  the  free  surface,  then  will 

P  =  Dgz\ •     .     (437) 

which  shows  that  2wesswcs  are  'proportional  to  depths. 

Denoting  by  b  the  area  of  the  surface  pressed,  and  by  db  the  element 
of  this  surface,  whose  co-ordinate  is  z,  we  have,  Equation  (307),  for  the 
•pressure  pt  upon  this  element, 

pt  ==  Dg.z.  db, 
and  the  same  for  any  other  element  of  the  surface  ;  whence,  denoting 
the  entire  pressure  by  P,  we  shall  have 

P  =  lp/  =  Bg.lz.db (437") 

But  if  z  denote  the  co-ordinate  of  the  centre  of  gravity  of  the  entire 
surface  b,  then  will,  Equations  (91),     Zz.db  =  bzt,     and 

P=zDg.b.zi (438) 

Now  bzt  is  the  volume  of  a  right  cylinder  or  prism,  whose  base  is  b  and 
altitude  Zf\  Dg.b.zt  is  the  weight  of  this  volume  of  the  pressing  fluid. 
Whence  we  conclude,  that  the  pressure  exerted  upon  any  surface  by  a 
heavy  fluid  is  equal  to  the  weight  of  a  cylindrical  or  prismatic  column  of 
the  fluid  whose  base  is  equal  to  the  surface  pressed,  and  whose  altitude  is 
equal  to  the  distance  of  the  centre  of  gravity  of  the  surface  below  the  upper 
surface  of  the  fluid. 

When  the  surface  pressed  is  horizontal,  its  centre  of  gravity  will  be 
at  a  distance  from  the  upper  surface  equal  to  the  depth  of  the  fluid. 

This  result  is  wholly  independent  of  the  quantity  of  the  pressing 
fluid,  and  depends  solely  upon  the  density  of  the  fluid,  its  height,  and 
the  extent  of  the  surface  pressed. 


Example  1.  —  Required  the  pressure 
.against  the  inner  surface  of  a  cubical  ves- 
sel  filled  with  water,  one  of  its  faces  being 
horizontal.  Call  the  edge  of  the  cube  a, 
the  area  of  each  face  will  be  a2,  the  dis- 
tance of  the  centre  of  gravity  of  each 
vertical   face    below  the    upper  surface   will    be   -J-rt,  and    that   of   the 


f\ 

\ 

— \ 

MECHANICS    OF    FLUIDS 


303 


lower    face   a ;      whence,    the    principle     of    the     centre    of    gravity 
gives. 

4  a2  X  I  a  +  a2  X  a 


5«2 


8  a 


Again, 


b  —  5  a2; 

and   these,  substituted  in   Equation   (438).  give 

P  -  D  .g-b.z,  =  B.g.Sa3. 

Now    D g  x   I3  =  Dg,  is  the  weight  of  a  cubic  foot  of  water  ^62,5 
lbs.,  whence, 


lbs. 

62,5  X  3a3. 


Make   a  =  7  feet,  then  will 


lbs. 


P  =  62,5  X  3  x  (7)3  =  64312,5. 

The  weight  of  the  water  in  the  vessel  is  62,5  a3,  yet  the  pressuw 
is  62,5  X  3a3,  whence  we  see  that  the  outward  pressure  to  break 
the   vessel,  is.  three  times   the    weight   of  the    fluid.  . 

Example  2. — Let  the  vessel  be  a  sphere  filled  with  mercury,  and 
let  its  radius  be  R.  Its  centre  of  gravity  is 
at  the  centre,  and  therefore  below  the  upper 
surface  at  the  distance  R.  The  surface  of  the 
sphere  being  equal  to  that  of  four  of  its 
great  .circles,  we  have 

b  =  4«R2', 

whence. 


and,  Equation  (438), 


b.tt  =  4*i23; 


P  =  4*.£.g.R3. 


The  quantity  Dg  X  V  =  D g,  is  the  weight  of  a  cubic  foot  of 
mercury  =843,75  lbs.,  and  therefore,  substituting  the  value  ( of 
r  =  3,1416, 


lbs. 


P  =  4  x  3,1416  x  843,75  .  R3. 


304 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


Now  suppose  the  radius  of  the  sphere  to  be  two  feet,  then  will 
R3  =  8,  and 

lbs.  lbs. 

P  =  4  x  3,1416  X  843,75  X  8  =  84822,4. 

The  volume  of  the  sphere  is  £  *  R3 ;  and  the  weight  of  the  con- 
tained mercury  will  therefore  be  ±*R3gD  —  W%  Dividing  the 
whole   pressure   by  this,  we  find 

whence  the   outward   pressure  is  three  times  the  weight  of  the  fluid. 

Example  3. — Let   the  vessel  be  a    cylinder,  of    which    the    radius 
r  of  the  base   is   2,    and    altitude    I,  6   feet.     Then    will 

b.z,  =  «rl{r  +  I)  =  3,1416  X  2  X  6  X  8; 

which,  substituted  in  Equation  (438), 

P  =  301,5936  X  Dg, 

*nd 

W  -  3,1416  x  22  x  6  X  Dg  =  75,398  x  Dg\ 

whence, 

P    _  301,5936  X  Dg 

W  ~      75,3984 .  D  g     ~~      * 

lhat  is,  the  pressure  against  this  particular  vessel  is  four  times  the 
Aeight   of  the   fluid. 

§  268.— The  point  through  which  the  resultant  of  the  pressure 
upon  all  the  elements  of  the  surface 
passes,  is  called  the  centre  of  pressure. 
i<et  E I F  be  any  plane,  and  MN 
(he  intersection  of  this  plane  produced 
with  the  upper  surface  of  the  fluid 
which  presses  against  it.  Denote  the 
area  of  any  elementary  portion  n  of 
the  plane  E  IF  by  db  ;  and  let  m  be 
the  projection  of  its  place  upon  the 
upper  suiface  of  the  fluid;  draw  m  M 
perpendicular  to  M N,  and  join  n  with   Af  by    the   right    line  n  M,  th* 


MECHANICS    OF    FIT!  IDS.  305 

latter   will   also  be  perpendicular  to  M  N,  and    the  angle  nAfin  will 
measure   the    inclination   of    the   plane   EIF  to    the    surface    of  the 
fluid.     Denote  this  angle  by  9,  the  distance  m  n  by  h\  and  Mn  by  r' 
then   will 

hT  =  r'sin<p; 
the   pressure   upon   the   element   db, 

D  g  .>'  sin  9  db\ 

its    moment   with   reference   to   the   line  MN, 

D gr'2  sin  9  .  db\ 

and  for    the   entire  surface,  the   moment   becomes 

D  g .  sin  9  .  2  r'2  d  b. 

Denote  by  r  the  distance  of  the  centre  of  gravity  of  the  surface 
pressed  from  the  line  M  N,  its  distance  below  the  upper  surface  of 
the  fluid  will  be  r .  sin  9 ;  and  the  pressure  upon  this  surface  will  be  ' 

D  g .  r  sin  9  .  b  ; 

and  if  /  denote  the  distance  of  the  centre  of  pressure  from  the 
line  M N,  then  will 

Dg  .ramp.b.l  =  Dg  .  sin  9  .  2  r'2 .  db, 
from  which  we  have, 

Irt  ^_7±=.t_±I.i (439) 

r .  o  r  ,   •  . 

whence,  Equation  (238),  the  centre    of  pressure    is  found  at  the   centre 
cf  percussion  of  the  surface  pressed. 

The  principles  which  have  just  been  explained  are  of  grtirit 
practical  importance.  It  is  often  necessary  to  know  the  precise 
amount  of  pressure  exerted  by  fluids  against  the  sides  of  vessels 
and  obstacles  exposed  to  their  action,  to  enable  us  so  to  adjust 
the  dimensions  of  the  latter  as  to  give  them  sufficient  strength  to 
resist.  Reservoirs  in  which  considerable  quantities  of  water  are  col- 
lected and  retained  till  needed  for  purposes  of  irrigation,  the  supply 
of   cities   and   towns,  or  to   drive    machinery ;    dykes   to   keep  the  sea 


306 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


and  lakes  from  inundating  low  districts ;  artificial  embankments  con- 
structed along  the  shores  of  rivers  to  protect  the  adjacent  country 
in  times  of  freshets ;  boilers  in  which  elastic  vapors  are  pent  up  in 
a  high  state  of  tension  to  propel  boats  and  cars,  and  to  give  motion 
to   machinery,  are  examples. 

§  269. — As  a  single  instance,  let  it  be  required  to  find  the  thick 
ness  of  a  pipe  of  any  material  necessary  to  resist  a  given  pres- 
sure. 

Let  A  B  C  be  a  section  of  pipe  perpen- 
dicular to  the  axis,  the  inner  surface  of 
which  is  subjected  to  a  pressure  of  p  pounds 
on  each  superficial  unit.  Denote  by  R  the 
radius  of  the  interior  circle,  and  by  /  the 
length  of  the  pipe  parallel  to  the  axis ; 
then  will  the  surface  pressed  be  measured 
by  %  f  R .  / ;  and  the  whole  pressure  by 
2*R.l.p. 

By  virtue  of  the  pressure,  the  pipe  will  stretch ;  its  radius  will 
become  R  -f  d  R,  the  path  described  by  the  pressure  will  be  d  R, 
and   its   quantity  of  work 

2«  R.l.pdR. 

The  interior  circumference  before  the  pressure  was  2tfi£,  afterwards 
2<k{jR  -j-  dR),  and  the  path  described  by  resistance,  2tfdR.  And 
if  B  denote  the  resistance  which  the  material  of  the  pipe  is  capable 
of  opposing,  to  a  stretching  force,  without  losing  its  elasticity  over 
each  unit  of  section,  t  the  thickness  of  the  pipe,  then,  by  the  prin- 
ciple of  the   transmission  of  work,  must 


whence, 


2«.B.l.dR.t  =  2<xR.l.p.dR; 


Rp 
B 


The   value  of  p   is   estimated    in   the   case   of  water   pressure   by 
the, rules  just  given.     That  in  the  case  of  steam  or  condensed  gases, 


MECHANICS     OF    FLUIDS. 


,307 


by  rules  to  be  given  presently.  The  7alue  of  B  is  readily  obtained 
from  Taole  I,  giving  the  results  of  experiments  on  the  strength  of 
materials. 


EQUILIBRIUM   AND   STABILITY    OF   FLOATING    BODIE8. 

g  270. — When  a  body  is  immersed  in  a  fluid  it  is  not  only 
acted  upon  by  its  own  weight,  but  also  by  the  pressure  arising  from 
the  weight  of  the  fluid,  and  the  circumstances  of  its  rest  or  motion 
will  be    made    known    by   Equations  (B)  and  (C). 

Let  ED  be  the  body  ;  take  the  plane  x  y  in  the  plane  of  the  up. 
per  surface  of  the  fluid,  ,   ; 

supposed  at  rest,  and 
the  axis  of  z  therefore 
vertical.  Denote  by 
b  the  entire  surface 
of  the  body,  and  by 
d  b,  one  of  its  elements, 
whose  co-ordinates  of 
position  are  x  y  z.  The 
pressure  upon  this  ele- 
ment will  be  - 

D.g.z.db, 

in  which  D  is  the  density  of  the  fluid,  and  g  the  force  of  gravity. 

This  pressure  is,  §  248,  normal  to  the  surface,  and  denoting  by 
a,  f3  and  7,  the  angles  which  this  normal  makes  with  the  axes  xyz, 
respectively,  the  components  of  the  pressure  in  the  direction  of  these 
axes  will  be 

D  •  g  .z  .db  .  cos  a  ;     D.g.z.db.  cos  (3  ;     D.g.z.db.  cos  y. 

Similar  expressions  being  found  for  the  components  of  the  pressure  on 
other  elements,  we  have,  by  taking  their  sum, 

Dg.2  z.db  .cos  a;     Dg .  2  z  .  db  .  cos/3  ;     Dg .  2  z  .  db  .  cos  y. 

But  rffi.cosa,    (/6.COS/S,    and    db.cosy,    are    the  projections   of  the 
area  db  on  the  co-ordinate  planes  z  y,  z  x  and  x  y,  respectively;    and 


308  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

2  z  .  d  b  .  cos  a,  2  z.  d b  .  cos  /3,  2z.d  b  .  cos  7,    are    volumes    of    righ 
prisms    whose     bases   are     projections    of    the    entire    surface    pressed 
upon    the   same  co-ordinate  planes,  and  of  which  the  altitude  of  each 
ia  the  depth  of  the  common  centre  of  gravity  of  the   elements  of  its 
base  submerged  to  the  depths  of  their  corresponding  surface  elements. 

Whence  we  conclude,  that  the  component  of  the  pressure  on  ang 
surface,  estimated  in  any  direction,  is  equal  to  the  pressure  on  so  much 
of  that  surface  as  is  equal  to  its  projection  on  a  plane  at  right  angles 
to    the   given   direction. 

The  cylinder  or  prism  which  projects  an  element  on  one  side  of 
the  body  will  also  project  an  element  situated  on  the  opposite  side  ; 
these  projections  will,  therefore,  be  equal  in  extent,  but  will  have 
contrary  signs,  for  the  normal  to  the  one  will  make  an  acute,  and 
to  the  other  an  obtuse  angle  with  the  axis  of  the  plane  of  projection. 
When  these  projections  are  made  upon  any  vertical  plane,  the  value 
of  z  will  be  the  same  in  both,  and  hence,  for  each  positive  product, 
z  .db.  cos  a  and  z  .  db  .  cos /3,  there  will  be  an  equal  negative  product ; 
therefore, 

Dg.2z.db.cosa.  —  2  P  cos  a  =  0;  Dg  .  2  z  .db  .  cos  (3  =  2  Pcos  /3=0. 

That  is,  the  sum  of  the  horizontal  pressures  in  the  directions  of 
■x  and  y,  and  therefore  in  all  horizontal  directions,  will  be  zero  ;  and 
the   first   and   second   of  Equations  (120),  give 

d2  x         .  d?y 

or,  which  is  the  same  thing,  there  can  be  no  horizontal  motion  of 
translation    from    the   fluid   pressure. 

When  the  projections  of  opposite  elements  are  made  upon  a 
Horizontal  plane,  they  will  still  be  equal  with  contrary  signs,  the 
normal  to  the  elements  on  the  lower  side  making  obtuse,  while  the 
normals  to  the  elements  above  make  acute  angles  with  the  axis  z ; 
tout  the  corresponding  values  of  z  will  differ,  and  by  a  length  equal 
to  that  of  the  vertical  filament  of  the  body  of  which  these  elements 
form    the   opposite   bases,  and   hence 

Dg.  2z .db .  cosy  =■  Dg .2  (z'—z4)d b cosy  =  —  Dg2c  i  6 cosy  ••  (440) 


MECHANICS    OF     FLUIDS.  309 

in  which  z'  denotes  the  ordinate  for  the  upper,  and  zt  that  for  the 
lower  element  in  the  same  vertical  line,  and  c  the  distance  between 
the  elements;  and   the   third  of  Equations  (120)  becomes 

(d2  z  \  dlz 

P  cos  y  —  m  •  -r-g"/  =  ^9  ~~  D  9  '^c'db*  cos  y  —  2  m  •  -——  =  0. 

But  1c .  db .  cos y  is  the  volume  of  the  immersed  body  which  is 
obviously  equal  to  that  of  the  displaced  fluid;  also  J)g  ,2  c  .d  b  .cosy 
is  the  wreight  of  the  displaced  fluid ;  and  Mg  that  of  the  body. 
Denoting  the  volume  of  the  body  by  V\  its  density  by  D\  the 
above   may  be  written 

d2z 
V'D'g  _  v'Dg  -2m.—  =  0.     •     •    .     (441) 

Now,  when 

V'D'g  -  V Dg  =  0, 


or 


D  =  D\ 


then  will 


^       d2z 

*m-dW  =  °> 

and    there   can    be   no    vertical  motion    of  translation    fi:m    the    fluid 
pressure  and   the   body's   weight. 
When  D'  >  2),  then  will 

d2z 
■Zm.J¥={D'-D)V,9i 

and    the   body  will    sink  with   an   accelerated   motion. 
When  D   <  D,  then  will 

2m~=-(D'-D)  V'.g, 

and    the   body  will  rise  with  an  accelerated  motion   till 

2m.°^  ^  V  D'g  -  VDv  =  0',     •     •  fcl42) 


310 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


in  which    V  denotes  the  volume  ABC,  of  the 
iluid  displaced.     At   this   instant  we  have 


V'D'g  =  VLg- 


(443) 


and  if  the  body  be  brought  to  rest,  it  will 
remain  so.  That  is,  the  body  will  float  at  the 
surface  when  the  weight  of  the  fluid  it  dis- 
places is  equal  to  its  own   weight. 

The  action  of  a  heavy  fluid  to  support  a  body  wholly  or  partly 
immersed  in  it,  is  called  the  buoyant  effort.  The  intensity  of  the 
buoyant  effort  is  equal    to  the  weight  of  the  fluid  displaced. 

Substituting  the  values  of  the  horizontal  and  vertical  components 
of  the   pressures  in  Equations  (118),  and  reducing  by  the   relations, 


Dg  .2  c  .  d  b  .  cos  y  .  x'  =  D  g  .  V.  x  ; 
D  g  .  2  c  .d  b  .  cos  y .  y'  =  D  g .  V .  y  ; 


(444) 


in  which  x  and  y  are  the  co-ordinates  of  the  centre  of  gravity  of  the 
displaced  fluid  referred  to  the  centre  of  gravity  of  the  body,  we  find 


x' .  d2y'  —  y' .  d2  xf 

*    .  „  =  0  ; 


2  m 


2  m 


d& 

6 

-d2 

x'  -  x' 

•  d2z' 

dt2 

y'' 

d2 

z'  -  z' 

d2y' 

dt2 


—  Dg*  V'X\ 
—  —  Dg-V-y. 


(445) 


Equations  (444)  show  that  the  line  of  direction  of  the  buoyant 
effort  passes  through  the  centre  of  gravity  of  the  displaced  fluid. 
This  point  is  called  the  centre  of  buoyancy.  And  from  Equations 
(445),  we  see  that  as  long  as  x  and  y  are  not  zero,  there  will  be 
an    angular  acceleration  about   the   centre  of  gravity.     At  the   instant 

x  =  0    and    y  =  0,  that   is   to    say,  when    the   centres   of  gravity  of 
the    body    and   displaced    fluid    are    on    the   same  .  vertical   line,    this 
acceleration   will    cease,    and    if    the   body    were   brought   to  rest,   it 
would   ha\e   no   tendency  to   rotate. 
To    recapitulate,  wre  rind, 


MECHANICS     OF    FLUIDS. 


311 


1st.  That  the  pressures  upon  the  surface  of  a  body  immersed  in 
a  heavy  fluid  have  a  sinyle  resultant,  called  the  buoyant  effort  of  the 
ffuid,  and    that    this  resultant    is    directed    vertically    upwards. 

2d.  That  the  buoyant  effort  is  equal  in  intensity  to  the  weight  of 
the  fluid   displaced. 

3d.  That  the  line  of  direction  of  the  buoyant  effort  passes  through 
the   centre    of  gravity  of  the   displaced  fluid. 

4  th.    That  the    horizontal  pressures   destroy    one    another. 

§271. — Having  discussed  the  equilibrium,  consider  next  the  sta 
bility  of  a  floating  body.  The  density  of  the  body  may  be  homo- 
geneous or  heterogeneous. 
Let  A  BCD  be  a  section 
of  the  body  by  the  upper 
surface  of  the  fluid  when 
the  body  is  at  rest,  G 
its  centre  of  gravity,  and 
It  that  of  the  fluid  dis- 
placed. Denote  by  V  the 
volume  of  the  displaced 
fluid,  and  by  M  the  mass 
of   the    entire   body.     The 

body  being  in  equilibrio,  the  line   ^JJ  will  be   vertical,  and  denoting 
the   density  of  the    fluid   by  D,  we    shall  have 


M  =  D.  V. 


(446) 


Suppose  the  section  ABCD  either  raised  above  or  depressed 
below  the  surface  of  the  fluid,  and  at  the  same  time  slightly  careened ; 
also  suppose,  when  the  body  is  abandoned,  that  the  elements  have 
a  slight  velocity  denoted  by  u,  «',  &c.  Now  the  question  of  sta- 
bility will  consist  in  ascertaining  whether  the  body  will  return  to  its 
former  position,  or  will    depart  more   and   more  from    it. 

The  free  surface  of  the  fluid  is  called  the  plane  of  floatation, 
and  during  the  motion  of  the  body  this  plane  will  cut  from  it  a 
variable  section. 

Let   A'  B'  6"  D'  be  one  of  these  sections   at  any  g'ven  instant  of 


312  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

time;  A  B"  C D",  another  variable  section  of  the  body  by  a  hori 
zontal  plane  through  the  centre  of  gravity  of  the  primitive  section 
A  BCD.  and  A  C  the  intersection  of  the  two  Denote  bv  0  the 
inclination  of  these  two  sections,  and  by  £  the  vertical  distance  of 
A  B"  CD",  from  the  plane  of  floatation,  which  now  coincides  with 
A'  B'  C  D\  this  distance  being  regarded  as  negative  or  positive,  ac- 
cording as  A  B"  C  D"  is  below  or  above  the  plane  of  floatation. 
The  variable  quantities  0  and  £  will  be  supposed  very  small  at  the 
instant  the  body  is  abandoned.  Will  they  continue  so  during  the 
whole  time  of  motion  ? 

From  the  principles  of  living  force  and  quantity  of  work,  we  have, 
Equation  (121), 

fu2.dM  =  2f(Xdx  -f  Ydy  4-  Zdz)  +  C. 

The  forces  acting  are  the  weights  of  the  elements  dM  and  the  verti- 
cal pressures,  the  horizontal  pressures  destroying  one  another ;  whence, 
X  =  0,     Y  =  0,    and 

JutdM  =  %fz  dz  +  C=22Zz  -f  C      .     .      (447) 

The  force  which  acts  upon  the  body  downwards  is  its  own 
weight,  and  the  force  which  acts  upon  it  upwards  is  the  difference 
between  its  own  weight  and  that  of  the  fluid  it  displaces;  the  first 
will  be  the  integral  of  g.dM,  and  the  second  that  of  g.D.dV, 
whence, 

^Zz  —  Jg.z.dM  —  JgD.z.dV.      .     .     .     (448) 

But,  drawing  from  the  centre  of  gravity  G,  of  the  body,  the  perpen- 
dicular G  E,  to  the  plane  of  floatation  A'  B'  C  D' ,  and  denoting  G  E 
by  zt,  we  have 

■ 

/  g  .  z  .d  M  =  g  Mzr 

The  integral  J  gD.  z.  d  V,  will  be  divided  into  two  parts,  viz:  one 
relating  to  the  volume  of  the  body  below  A  B  CD,  or  the  volume 
immersed  in  a  state  of  rest,  and  the  other    that    comprised   between 


MECHANICS    OF     FLUIDS.  313 

A  BCD  and  the  plane  of  floatation  A'  B'  C  D',  when  the  body  is  in 
motion.  Denote  by  g  D  V  z\  the  value  of  the  first,  in  which  zf 
denotes  the  variable  distance  H F,  of  the  centre  of  gravity  H,  of 
the  volume  V,  from  the  plane  of  floatation  A'  B'  C  Df.  And  repre- 
senting for  the  instant  by  h  tne  value  of  the  integral  /  zdV,  com- 
prehended between  the  planes  A  BCD  and  A'  B'  Cf  Df,  gDh  will 
be    the   second    part;  and  Equation  (447)  becomes 


f 


utdM  =  2g.Mzt  -  2gDVz'  -  2gDh  +  C.  •  •  (449) 

The  line  G  //,  being  perpendicular  to  the  plane  A  B  C  D,  the  angle 
which  it  makes  with  the  line  G  E  is  equal  to  d,  and  denoting  the  dis- 
tance  G  H  by  ay  we  have 

zt  —  zr  ±  a  cos  8  ; 

the  upper  sign  being  taken  when  the  point  G  is  below  the  point 
II,  and  the  lower  when  it  is  above.  This  value  reduces  Equation 
(449)    to 

fu2dM  =  ±2gD  Vacosd  —  2gDk  +  C.    •    •   •  (450) 

Let  us  now  find  the  integral  //.  For  this  purpose,  conceive  the 
area  ABCD  to  be  divided  into  indefinitely  small  elements  denoted 
by  d\  and  let  these  be  projected  upon  the  plane  of  floatation. 
A'  B'  C  D'.  The  projecting  surfaces  will  divide  the  volume  com 
prised  between  these  two  sections  into  an  indefinite  number  of 
vertical  elementary  prisms,  and  these  being  cut  by  a  series  of  hori- 
zontal planes  indefinitely  near  each  other,  will  give  a  series  of  ele- 
mentary volumesr  each  of  whicji  will  be  denoted  by  d  V}  and  we 
shall    have 

d  V  =  dz  .  d\.  cosd  ; 

whence,  for  a    single  elementary   vertical    prism 

JzdV  —  Jzdz.dX.cosQ  =  J  (z)2  .  cts  0  .  d\ ; 
in  which  (z)  denotes  the  mean  altitude  of  the  prism,  and  consequently 

h  =  I  cosd.  f{zf.d\ 
which   must    be    extended    to   embrace    the    entire    sur  ace  A  B  CD, 


314  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

The  value  of  (z^  is  composed  <  f  two  parts,  viz.  :  one  comprised 
between  the  parallel  sections  A' B' C  D'  and  AB"CD",  and  which 
has  been  denoted  by  £;  the  other  comprised  between  the  base  d\ 
and  the  second  of  these  planes,  and  which  is  equal  to  / ,  sin  <?,  de- 
noting by  I  the    distance  of  dX  from    the  intersection  A  C\    whence, 

(e)  =  I  +  J.  sin*, 

ir,  which  I  will  be  positive  or  negative  according  as  d\  happens  to 
be  below  or  above  the  plane  A  B"  C  D".  Substituting  this  in  the 
value  of  h,  and  recollecting  that  £  and  &  are  constant  in  the  inte- 
gration, we   find  » 

h  =  $%*.  cos  d  .  f  dX  +  ^sind  cost  fld\  +  \  sin2d  .  cost  fl?d\. 

I  r 

Denote    by    b    the   area  of  A  B  C  D,  or    the   value   of   /  d  X.     The 

line  A  C  passing  through  the  centre  of  gravity  of  A  B  C D,   we  have 
/  Idk  =  0.     And  denoting  by  Jfc    the  principal    radius  of   gyration  of 
the  surface  b,  in  reference  to  the  axis  AC,  its  moment  of  inertia  is 

fPdk  =  bk% 

in  which  the  value  of  ht  is  dependent  upon  the  figure  and  extent 
of  the  surface  A  BCD,  and  upon  the  position  of  the  line  AC. 
Whence, 

h  =  \  b  .  cos  6  (£2  +  k*  sin2  0).      ....     (451 ) 

Taking 

sin  6^6  —  — -  -f  &c  ;  ,  cos  &  =  1  —  _  -f  &c. 

Neglecting  all  the  terms  of  the  third  and  higher  orders,  substitut- 
ing in  the  value  of  h,  and  then  in  Equation  (450)  we  find,  after  trans- 
posing and    including   the   term    ±2g  D  Va,  in  the  constant  C, 

fuKdM+  gD^b?  +  (bk*  ±  Va)6*\=  C.  .  .  .(452) 

Now  the  value  of  the  constant  C  depends  upon  the  initial  values 
of  u,  6  and  £ ;  but  these  by  hypothesis  are  very  small ;  hence  (7, 
must  also   be  very   small.     As  long  as  the  second  term  cf  the  first 


MECHANICS    OF    FLUIDS.  315 

member  is  positive,  /  ti2dM  must  remain  very  small,  since  it  is  essen- 
tially positive  itself,  and  being  increased  by  a  positive  quantity, 
the  sum  is  very  small.  Hence  £  and  0  must  remain  very  small. 
But    when    the    second   term   is   negative,    which    can    only    be    when 

bk2  ±Va    is   negative  and  greater   than    6— ,    the  value    of     I  ifidM 

may  increase  indefinitely  ;  for,  being  diminished  by  a  quantity  that 
increases  as  fast  as  itself,  the  difference  may  be  constant  and  very  small. 
Hence,  C,  and  6  may  increase  more  and  more  after  the  body  is  abandoned 
to  itself,  and  finally  it  may  overturn. 

The  stability  of  the  equilibrium  depends,  therefore,  upon  the  sign  of 
bk2  ±Va;    the    equilibrium    is   always   stable    when   this    quantity    is 

<T2 

positive;    it   is   unstable   when   it   is  negative   and   greater   than    b—. 

The   value    of   bk2  =  I  PdX,   must  always   be    positive,    since    all    its 

elements  are  positive ;  the  value  of  ±  Va  becomes  negative  when  the 
centre  of  gravity  of  the  body  is  above  that  of  the  displaced  fluid,  in 
which  case  the  stability  requires  that  the  moment  of  inertia 

bk2yVa, 

When  the  centre  of  gravity  of  the  body  is  below  that  of  the  displaced 
fluid,  the  sign  of  Va  is  positive. 

Whence  we  conclude  that  the  equilibrium  of  a  body  floating  at  the 
surface  of  a  heavy  fluid  will  be  stable  as  long  as  the  centre  of  gravity  of 
the  body  is  below  that  of  the  displaced  fluid  ;  that  it  will  also  be  stable 
about  all  lines  AC,  with  reference  to  which  the  moment  of  inertia  of  the 
surface  ABCD  is  greater  than  Fa,  the  product  of  V  by  the  distance  a 
between  the  centres  G  and  H.  When  this  condition  is  not  fulfilled,  the 
equilibrium  may  be  unstable.  A  ship  whose  centre  of  gravity  is  above 
that  of  the  water  she  displaces,  may  overturn  about  her  longer,  but  not 
about  a  shorter  axis. 

§  272. — A  line  B K  through  the  centre  of  gravity  G  of  the  body 
and  which  is  vertical  when  the  body  is  in  equilibrio,  is  called  a  line  of 
rest.  A  vertical  line  H  M  through  the  centre  of  gravity  H'  of  the 
displaced  fluid  is  called  a  line  of  support.     The  point  M,  in  which  the 


316 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


line  of  support  cuts  the  line  of  rest  is  called  the  metacentre.     The  body 
will  be  in  equilibrio  when  the  line  of  rest  and  of  support  coincide.     The 
equilibrium    will   be   stable    if 
the  metacentre  fall  above  the 
centre  of  gravity;    it  may  be 
unstable  if  below. 


§  273.— When  the  equi- 
librium is  stable,  and  the  body 
is  disturbed  and  then  abandoned 
to  the  action  of  its  own  weight 
and  that  of  the  fluid  pressure, 

it  will,  in  its  efforts  to  regain  its  place  of  rest,  oscillate  about  this 
position,  and  finally  come  to  rest. 

All  changes  of  level  in  the  liquid  itself  are  disregarded  in  the  above 
discussion.  If  caused  by  the  floating  body,  the  liquid  will  rise  on  one 
side  and  sink  on  the  other ;  and  the  resistances  thus  developed  will  tend 
to  check  oscillation  and  increase  stability.     But  this  is  not  the  case  for 

m 

the  effects  of  waves  and  other  motions  in  the  liquid  upon  the  oscillating 
body.  The  rolling  of  a  ship  in  a  disturbed  sea  is  indeed  a  very  compli- 
cated problem  not  yet  solved. 


SPECIFIC     GRAVITY. 

§  2*74. — The  specific  gravity  of  a  body  is  the  weight  of  so  much  of 
the  body  as  would  be  contained  under  a  unit  of  volume. 

It  is  measured  by  the  quotient  arising  from  dividing  the  weight  of 
the  body  by  the  weight  of  an  equal  volume  of  some  other  substance, 
assumed  as  a  standard ;  for  the  ratio  of  the  weights  of  equal  volumes  of 
two  bodies  being  always  the  same,  if  the  unit  of  volume  of  each  be 
taken,  and  one  of  the  bodies  become  the  standard,  its  weight  wi'l  become 
the  unit  of  weight. 

The  term  density  denotes  the  degree  of  proximity  among  the 
particles  of  a  body.  Thus,  of  two  bodies,  that  will  have  the  greater 
density  which  contains,  under  an  equal  volume,  the  greater  number 
of  particles.     The  weight  of  a  substance  is  directly  proportional  to  its 


MECHANICS    OF    FLUIDS.  317 

density,  and  the  ratio  of  the  weights  of  equal  volumes  of  two  bodies 
is  equal  to  the  ratio  of  their  densities.  Denote  the  weight  of  the  first 
by  W,  its  density  by  D,  its  volume  by  V,  and  the  force  of  gravity 
by  #,  then  will 

and  denoting  the  like  elements  of  the  othei  body  by  Wf ,  Dt  and 
V t ,  we   have 

w,=9.d,.v,. 

Dividing   the   first   by  the  second, 

W         gDV  BV 


W, 

7  9»4 

~ 

I>, 

y, ' 

and 

making 

the 

volumes 

equal, 
W 

w, 

m 

J) 

(453) 

Now  suppose  the  body  whose  weight  is  W{  to  be  assumed  as  the 
standard  both  for  specific  gravity  and  density,  then  will  D{  be  unity, 
and 

in  which  S  denotes  the  specific  gravity  of  the  body  whose  density 
is  D ;  and  from  which  we  see,  that  when  specific  gravities  and 
densities  are  referred  to  the  same  substance  as  a  standard,  the 
numbers  which   express    the   one  will    also   express   the    other. 

§275. — Bodies  present  themselves  under  every  variety  of  condi- 
tion—gaseous, liquid  and  solid;  and  in  every  kind  of  shape  and  of 
all  sizes.  The  determination  of  their  specific  gravity,  in  every  in- 
stance, depends  upon  our  ability  to  find  the  weight  of  an  equal 
volume  of  the  standard.  When  a  solid  is  immersed  in  a  fluid,  it 
loses  a  portion  of  its  weight  equal  to  that  of  the  displaced  fluid. 
The  volume  of  the  body  and  that  of  the  displaced  fluid  are  equal. 
Hence  the1-  weight  of  the  body  in  vacuo,  divided  by  its  loss  of 
weight  when   immersed,  will    give    the    ratio  of  the  weights  of  equal 

volumes  of  the   body  and  fluid  ;   and    if  the    latter    be    taken    as   the 

"20 


318  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

standard,  and  the  loss  of  weight  be  made  to  occupy  the  denomi 
nator,  this  ratio  becomes  the  measure  of  the  specific  gravity  of  the 
bcdv  immersed.  For  this  reason,  and  in  view  of  the  consideration 
that  it  may  be  obtained  pure  at  all  times  and  places,  water  is 
assumed  as  the  general  standard  of  specific  gravities  and  densities 
for  all  bodies.  Sometimes  the  gases  and  vapors  are  referred  to 
atmospheric  air,  but  the  specific  gravity  of  the  latter  being  known 
as  referred  to  water,  it  is  very  easy,  as  we  shall  presently  see,  to 
pass  from  the  numbers  which  relate  to  one  standard  to  those  that 
refer    to    the    other. 

§  276. — But  water,  like  all  other  substances,  changes  its  density  with 
its  temperature,  and,  in  consequence,  is  not  an  invariable  standard. 
It  is  hence  necessary  either  to  employ  it  at  a  constant  temperature, 
or  to  have  the  means  of  reducing  the  apparent  specific  gravities,  as 
determined  by  means  of  it  at  different  temperatures,  to  what  they 
would  have  been  if  the  water  had  been  at  the  standard  temperature. 
The  former  is  generally  impracticable;    the  latter  is  easy. 

Let  D  denote  the  density  of  any  solid,  and  S  its  specific  gravity, 
as  determined  at  a  standard  temperature  corresponding  to  which  the 
density  of  the  water  is  Dr     Then,  Equation  (453), 

Again,  if  S'  denote  the  specific  gravity  of  the  same  body,  as  indi- 
cated by  the  water  when  at  a  temperature  different  from  the  stan- 
dard, and  corresponding  to  which  it  has  a  density   Din  then  will 

D 


S'  = 


A, 


Dividing  the  first  of  these  equations  by  the  second,  we  have 

S'         D/ 
whence, 

S=  S'-^-, (455) 

and   if  the    density  Dt  ,  be  taken  as  unity, 
,,  S  =  S'D„.      -  .  (450) 


MECHANICS     OF    FLUIDS. 


319 


That  is  to  say,  the  specific  gravity  of  a  body  as  determined  at  tht 
standard  temperature  of  the  water,  is  equal  to  its  specific  gravity  deter- 
mined  at  any  other  temperature,  multiplied  by  the  density  of  the 
water  corresponding  to  this  temperature,  the  density  at  the  standard 
temperature    being  regarded  as  unity. 

To  make  this  rule  practicable,  it  becomes  necessary  to  find  tho 
relative  densities  of  water  at  different  temperatures.  For  this  pur- 
pose, take  any  pure  metal,  say  silver,  that  easily  resists  the  chemical 
action  of  water,  and  whose  rate  of  expansion  for  each  degree  of  Fahr. 
thermometer  is  accurately  known  from  experiment ;  give  it  the  form 
of  a  slender  cylinder,  that  it  may  readily  conform  to  the  temperature 
of  the  water  when  immersed.  Let  the  length  of  the  cylinder  at  the 
temperature  of  32°  Fahr.  be  denoted  by  /,  and  the  radius  of  its  base 
by  nd\    its  volume  at  this  temperature  will  be, 

77  m2  P  x  I  =  77  m?  Is. 

Let  nl  be  the  mean  expansion  in  length  for  each  degree  of  the  ther- 
mometer above  32°.  Then,  for  a  temperature  denoted  by  t,  will  the 
whole  expansion  in  length  be 

nl  x  (t  -  32°), 

and  the  entire  length  of  the  cylin- 
der   will    become 

l+nl(t-32°)  =  l[\+n  (*-32°)]; 

which,  substituted  for  I  in  the  first 
expression,  will  give  the  volume 
for  the  temperature  t,  equal  to 

«m2P[l  +  n(t  —  32°)]3. 

The  cylinder  is  now  weighed  in 
vacuo  and  in  the  wrater,  at  differ- 
ent temperatures,  varying  from  32° 
upward,  through  any  desirable  range, 
say  to  one  hundred  degrees.  The 
temperature  at  each  process  being 
substituted  above,  gives  the  volume 
of  the    displaced    fluid :    the    weight   of    the   displaced   fluid    is   known 


320       ELEMENTS    OF    ANALYTICAL    MECHANICS. 

from  the  loss  of  weight  of  the  cylinder.  Dividing  this  weight  by 
the  volume  gives  the  weight  of  the  unit  of  volume  of  the  water  at 
the  temperature  t.  It  was  found  by  Stampfer  that  the  weight  of 
the  unit  of  volume  is  greatest  when  the  temperature  is  38°. 75  Fah- 
renheit's scale.  Taking  the  density  of  the  water  at  this  temperature  as 
unity,  and  dividing  the  weight  of  the  unit  of  volume  at  each  of  the 
other  temperatures  by  the  weight  of  the  unit  of  volume  at  this, 
38°.75,  Table  II  will  result. 

The  column  under  the  head  V  will  enable  us  to  determine  how 
much  the  volume  of  any  mass  of  water,  at  a  temperature  /,  exceeds 
that  of  the  same  mass  at  its  maximum  density.  For  this  purpose, 
we  have  but  to  multiply  the  volume  at  the  maximum  density  by  the 
tabular  number  corresponding  to  the  given  temperature. 


ATMOSPHERIC     PRESSURE. 

§  277. — The  atmosphere  encases,  as  it  were,  the  whole  earth.  It 
has  weight,  else  the  expansive  action  among  its  own  particles  would 
cause  it  to  extend  itself  through  space.  The  weight  of  the  upper 
stratum  of  the  atmosphere  is  in  equilibrio  with  the  expansive  action 
of  the  strata  below  it,  and  this  condition  determines  the  exterior 
limit. 


Since  the  atmosphere  has  weight,  it  must 
exert  a  pressure  upon  all  bodies  within  it. 
To  illustrate,  fill  with  mercury  a  glass  tube, 
about  32  or  33  inches  long,  and  closed  at 
one  end  by  an  iron  stop-cock.  Close  the 
open  end  by  pressing  the  finger  against  it, 
and  invert  the  tube  in  a  basin  of  mercury; 
remove  the  finger,  the  mercury  will  not 
escape,    but   remain    apparently    suspended,    at 


MECHANICS     OF    FLUIDS.  321 

the   level   of  the  ocean,  nearly    30    inches   above    the    surface    cf  the 
mercury  in  the   basin. 

The  atmospheric  air  presses  on  the  mercury  with  a  force  sufficient 
to  maintain  the  quicksilver  in  the  tube  at  a  height  of  nearly  30 
inches  ;  whence,  the  intensity  of  its  pressure  must  be  equal  to  the  weight 
of  a  column  of  mercury  whose  base  is  equal  to  that  of  the  surface 
pressed  and  whose  altitude  is  about  30  inches.  The  force  thus  exerted. 
is  called  the  atmospheric  pressure. 

As  the  absolute  amount  of  atmospheric  pressure  was  first  discovered 
by  Torricelli,  the  tubes  employed  iu  such  experiments  arc  called 
Torricellian  tubjs,  and  the  vacant  space  above  the  mercury  in  the 
tube  is  called  the   Torricellian  vacuum. 

The  pressure  of  the  atmosphere  at  the  level  of  the  sea,  support- 
ing as  it  does  a  column  of  mercury  30  inches  high,  if  we  suppose 
the  bore  of  the  tube  to  have  a  cross-section  of  one  square  inch 
the  atmospheric  pressure  up  the  tube  will  be  exerted  upon  this 
extent  of  surface,  and  will  support  30  cubic  inches  of  mercury. 
Each  cubic  inch  of  mercury  weighs  0,49  of  a  pound — say  half  a 
pound — from  which  it  is  apparent  that  the  surfaces  of  all  bodies,  at 
the  level  of  the  sea,  are  subjected  to  an  atmospheric  pressure  of  fifteen 
pounds   to  each  square  inch. 

BAROMETER. 

§278. — The  atmosphere  being  a  heavy  and  elastic  fluid,  is  com- 
pressed by  its  own  weight.  Its  density  cannot  be  the  same  through- 
out, but  diminishes  as  we  approach  its  upper  limit  where  it  is  least, 
being  greatest  at  the  surface  of  the  earth.  If  a  vessel  filled  with 
air  be  closed  at  the  base  of  a  high  mountain  and  afterwards  opened 
on  its  summit,  the  air  will  rush  out ;  and  the  vessel  being  closed 
again  on  the  summit  and  opened  at  the  base  of  the  mountain,  the 
air  will   rush  in. 

The  evaporation  which  takes  place  from  large  bodies  of  water, 
the  activity  of  vegetable  and  animal  life,  as  well  as  vegetable 
decompositions,  throw  large  quantities  of  aqueous  vapor,  carbonic 
acid,    and    other   foreign    ingredients    temporarily    into    the    permanent 


322       ELEMENTS    OF    ANALYTICAL    MECHANICS 


portions  of  the  atmosphere.  These,  together  with  its  ever-varying 
temperature,  keep  the  density  and  elastic  force  of  the  air  in  a 
state  of  incessant  change.  These  changes  arc  indicated  by  the 
Barometer,  an  instrument  employed  to  measure  the  intensity  of 
atmospheric  pressure,  and  frequently  called  a  weather-glass,  because 
of  certain  agreements  found  to  exist  between  its  changes  and  those  of 
the  weather. 

The  barometer  consists  of  a  glass  tube  about  thirty-four  or  thirty- 
five  inches  long,  open  at  one  end,  partly  filled  with  distilled  mer- 
cury, and  inverted  in  a  small  cistern  also  containing  mercury.  A 
scale  of  equal  parts  is  cut  upon  a  slip  of  metal,  and  placed  against 
the  tube  to  measure  the  height  of  the  mercurial  column,  the  zero 
being  on  a  level  with  the  surface  of  the  mercury  in  the  cistern. 
The  elastic  force  of  the  air  acting  freely  upon  the  mercury  in  the 
cistern,  its  pressure  is  transmitted  to  the  interior  of  the  tube,  and 
sustains  a  column  of  mercury  whose  weight  it  is  just  sufficient  to 
counterbalance.  If  the  density  and  consequent  elastic 
force  of  the  air  be  increased,  the  column  of  mercury 
will  rise  till  it  attain  a  corresponding  increase  of 
weight;  if,  on  the  contrary,  the  density  of  the  air 
diminish,  the  column  will  fall  till  its  diminished 
weight  is  sufficient   to   restore    the    equilibrium. 

In    the    Common    Ba?ometer,  the    tube  and    its    cis- 
tern    are    partly    inclosed    in    a    metallic   case,    upon 
which  the  scale   is  cut,  the  cistern,  in  this  case,  hav- 
ing a  flexible  bottom,  against  which    a   plate  a  at  the 
end  of  a  screw  b  is  made  to  press,  in  order  to  elevate 
or  depress   the  mercury  in    the    cistern   to  the  zero  of 
the  scale, 

De  Luc's  Siphon  Barometer  consists  of  a  glass 
tube  bent  upward  so  as  to  form  two  unequal  par- 
allel legs :  the  longer  is  hermetically  sealed,  and 
constitutes  the  Torricellian  tube ;  the  shorter  is  open, 
and  on  the  surface  of  the  quicksilver  the  pressure 
of  the  atmosphere  is  exerted.  The  difference  be- 
tween   the   levels    in    the  longer   and   shorter   legs   is    the    barometric 


A 

\ 

31 

.— 

30 

SS£ 

29 

^1 

m 

*=f 

f 

r^5 

i 

b 

MECHANICS    OF    FLUIDS 


323 


r 


30 1=1 


height.     The   most  convenient  and  practicable  way  of  measuring  thi* 
difference,    is   to    adjust    a    movable    scale    between 
the   two   legs,    so    that   its   zero    may    be    made    to 
coincide    with    the    level    of    the    mercury    in     the 
shorter    leg. 

Different  contrivances  have  been  adopted  to  ren- 
der the  minute  variations  in  the  atmospheric  pres- 
sure, and  consequently  in  the  height  of  the  barome- 
ter, more  readily  perceptible  by  enlarging  the  di- 
visions on  the  scale,  all  of  which  devices  tend  to 
hinder  the  exact  measurement  of  the  length  of  the 
column.  Of  these  we  may  name  Morland's  Diago- 
nal, and  Hook's  Wheel-Barometer,  but  especially 
Huygcns1  Double-Barometer.  The  essential  properties 
of  a  good  barometer  are  perfection  of  vacuum,  width 
of  tube,  purity  of  the  mercury,  accurate  graduation  of 
the  scale,  and  a  good  vernier. 

§  279. — The  barometer  may  be  used  not  only  to  measure  the 
pressure  of  the  external  air,  but  also  to  determine  the  density  and 
elasticity  of  pent-up  gases  and  vapors.  When  thus  employed,  it  is 
called  the  barometer-gauge.  In  every  case  it  will 
only  be  necessary  to  establish  a  free  connection 
between  the  cistern  of  the  barometer  and  the  vessel 
containing  the  fluid  whose  elasticity  is  to  be  indi- 
cated ;  the  height  of  the  mercury  in  the  tube, 
expressed  in  inches,  reduced  to  a  standard  tempera- 
ture, and  multiplied  by  the  known  weight  of  a 
cubic  inch  of  mercury  at  that  temperature,  will 
give  the  pressure  in  pounds  on  each  square  inch. 
In  the  case  of  the  steam  in  the  boiler  of  an  en- 
gine, the  upper  end  of  the  tube  is  sometimes  left 
open.  The  cistern  A  is  a  steam-tight  vessel,  partly 
tilled7  with  mercury,  a  is  a  tube  communicating 
with  the  boiler,  and  through  which  the  steam  flows 
and  presses  upon  the  mercury  ;  the  barometer  tube 
be,    op«n    at    top,    reaches    nearly    to    the   bottom    of   the    vessel    A. 


60 


45 


30 


15 


324 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


having  attached  to  it  a  scale  whose  zero  coincides  with  the  level 
of  the  quicksilver.  On  the  right  is  marked  a  scale  of  inches,  and 
on   the  left    a    scale   of  atmospheres. 

If  a  very  high  pressure  were  exerted,   one  of  several    atmospheres 
for    example,    an    apparatus    thus    constructed    would 
require   a   tube   of   great   length,    in    which   case   Ma- 
riotte's  manometer  is   considered   preferable.     The  tube 
being  filled  with   air    and   the   upper   end   closed,    the 
surface    of  the   mercury  in   both   branches  will    stand 
at   the    same   level   as   long   as  no  steam  is  admitted. 
The    steam    being   admitted  through  d,  presses  on  the 
surface  of  the   mercury  a  and  forces  it  up  the  branch 
b  c,   and   the   scale    from    J   to   c   marks    the    force    of 
compression    in    atmospheres.      The   greater   width    of 
tube    is   given    at    a,    in   order  that   the   level   of  the 
mercury  at  this  point  may  not  be  materially  affected 
by  its   ascent  up    the   branch    be,  the   point  a    being   the   zero  of  the 
scale. 


§  280. — Another  very  important  use  of  the  barometer,  is  to  find 
the  difference  of  level  between  two  places  on  the  earth's  surface,  as 
the  foot   and    top    of  a   hill    or   mountain. 

Since  the  altitude  of  the  barometer  depends  on  the  pressure  of 
the  atmosphere,  and  as  this  force  depends  upon  the  height  of  the 
pressing  column,  a  shorter  column  will  exert  a  less  pressure  than  a 
longer  one.  The  quicksilver  in  the  barometer  falls  when  the  instru- 
ment is  carried  from  the  foot  to  the  top  of  a  mountain,  and  rises 
again  when  restored  to  its  first  position :  if  taken  down  the  shaft 
of  a  mine,  the  barometric  column  rises  to  a  still  greater  height.  At 
the  foot  of  the  mountain  the  whole  column  of  the  atmosphere,  from 
its  utmost  limits,  presses  with  its  entire  weight  on  the  mercury ; 
at  the  top  of  the  mountain  this  weight  is  diminished  by  that  of 
the  intervening  stratum  between  the  two  stations,  and  a  shorter 
column  of  mercury   will   be    sustained   by    it. 

It  is  well  known   that    the   surface   of    the    earth    is    not    uniform, 
and  does  not.  in  consequence,  sustain   an  equal   atmospheric   pressure 


MECHANICS    OF     FLUIDS.  325 

at  its  different  points;  whence  the  mean  altitude  of  the  barometric 
column  will  vary  at  different  places.  This  furnishes  one  of  the 
best  and  most  expeditious  means  of  getting  a  profile  of  an  extended 
section  of  the  earth's  surface,  and  makes  the  barometer  an  instru- 
ment of  great  value  in  the  hands  of  the  traveller  in  search  of 
geographical   information. 

§  281. — To   find   the   relation   which   subsists  between  the  altitudes 
of  two  barometric  columns,  and  the  difference  of  level  of  the  points 
where  they  exist,  resume  Equation  (427).     The  only   extraneous  force 
acting    being  that   of   gravity,    we   have,    taking   the   axis   z   vertical,, 
and  counting  z   positive   upwards, 

X  =  0;     Y  =  0;    Z=  -  g. 
and   hence, 

p  =  Ce~-ST (462) 

Making  2  =  0,  and  denoting  the  corresponding  pressure  by  p{,  we  find 

p,=  C; 

and   dividing  the  last    equation  by  this  one, 


P 


-  e     />, 


whence,  denoting   the  reciprocal  of  the   common  modulus  by  M, 

MP   ,       P, 
z  = log  — (463) 

9  P  V       ' 

Denote  by  h{  and  A,  the  barometric  heights  at  the  lower  and  upper 
stations,  respectively,  then  will 

and  reducing  the  barometric  column  h  to  what  it  would  have  been 
had  the  temperature  of  the  mercury  at  the  upper  not  differed  from 
that   at   the   lower  station,  by  Equation  (394),  we  have 

2i h 

p         h  [1  +  (T  -  T)  .0,0001001]' 

in  which   T  denotes  the  temperature  of  the  mercury  at  '.he  lowrr  and 

T'  that  at  the  upper  station. 

21 


326  ELEMENTS    OF    ANAL7TICAL    MECHANICS. 

Moreover,  Equation  (381), 

a  =  g'  (1  -  0,002551  cos  2  +)  ; 
in  which. 

f*  =  32,1808  =  force   of  gravity  at  the  latitude  of  45°. 

P 

Substituting    the  value  of  — -■,    of  <?,  and  that  of  P,  as  given    bj 

Equation  (393),  in  Equation   (463),  we  find 

MDmh„    l  +  (/-32°)0,00204     x     \K  1  "1 

•       D;     '  1-0,002551  cos 2+ X  °gLX  X  l+(T-  7")0,0001001  J' 

In  this  it  will  be  remembered  that  t  denotes  the  temperature  of 
the  air ;  but  this  may  not  be,  indeed  scarcely  ever  is,  the  same  at 
both  stations,  and  thence  arises  a  difficulty  in  applying  the  formula. 
But  if  we  represent,  for  a  moment,  the  entire  factor  of  the  second 
member,  into  which  the  factor  involving  t  is  multiplied,  by  X,  then 
we   may  write 

2=[l  +  (<-  32°)0,00204]  X. 

If  the  temperature  of  the  lower  station  be  denoted  by  t4 ,  and  this 
temperature  be  the  same  throughout  to  the  upper  station,  then  will 

f/  ie  [1  4.  {tf  -  32°)  0,00204]  X. 

And  if  the  actual  temperature  of  the  upper  station  be  denoted  by  /', 
and  this  be  supposed  to  extend  to  the  lower  station,  then   would 

z'  =  [1  +  (t*  -  32°)  0,00204]  X. 

Now  if  tt  be  greater  than  t\  which  is  usually  the  case,  then  will  the 
barometric  column,  or  A,  at  the  upper  station,  be  greater  than  would 
result  from  the  temperature  t\  since  the  air  being  more  expanded, 
a  portion  which  is  actually  below  would  pass  above  the  upper 
station  and  press  upon  the  mercury  in  the  cistern  ;  and  because  /* 
enters  the  denominator  of  the  value  X,  zt  would  be  too  small. 
Again,  by  supposing  the  temperature  the  same  as  that  at  the  uppei 
station  throughout,  then  would  the  air  be  more  condensed  at  the 
lower  station,  a  portion  of  the  air  would  sink  below  the  upper 
station  that  before  was  above  it,  and  would  cease  to  act  upon  the 
mercurial   column  h%  which  would,  in  consequence,  become  too  small' 


MECHANICS    OF    FLUIDS.  327 

and   this   would    make  zf   too   great.     Taking  a  mean   between   r    and 
z'  as  the   true    value,   we  find 

z  =  *'  *      =*  [If  I  (', +  V  -r-  64°)   0,00204]  X 


Replacing  X  by   its  value, 

Lx  x  F+(7T-  r]M6o7ooT  J 


i/"Z>,A,   l4-|(//  +  ''-G4°)0,00204  ("A,  1 

xlog   -^-x 


~     Dt  1  -0,002551  eos  2  + 

The  factor  — — --•>    we   have   seen,  is    constant,  and    it   only  rc- 

Dt 

mains    to    determine    its    value.       For    this    purpose,    measure    with 

aecuracv    the    difference    of   level    between    two    stations,    one    at    the 

base    and    the    other    on    the    summit    of    some   lofty    mountain,    bv 

means    of   a     theodolite,    or   levelling    instrument — this    will  give    the 

value    of  z ;     observe    the   barometric    column    at    both    stations — this 

will    give  h    and    ht  ;    take    also    the    temperature   of  the    mercury    at 

the     two     stations — this    will    give    T   and     T ;     and    by    a    detached 

thermometer     in     the     shade,     at     both    stations,    find     the    values    of 

tt  and  V .     These,  and    the  latitude  of  the    place,  being  substituted  in 

the    formula,    every    thing    will    be     known    except    the    co-efficient    in 

question,  which  may,  therefore,  be  found   by  the  solution  of  a  simple 

equation.     In    this    way,  it    is    found    that 

M  D„  h 


m  '  a 


-  60345,51   English  feet ; 


D, 

which  will  finally  give  for  z, 

ft.       i-f-i(;/  +  ;'_64°)0.00204  Vh      1  "1 

*=60345,oL     l_  o,002551  cos  2+    X  ,0gU~Xl  +( ^-7^)0,000 1001 J 

To  find  the  difference  of  level  between  any  two  stations,  the  lati- 
tude of  the  locality  must  be  known;  it  will  then  only  be  necessary 
to  note  the  barometric  columns,  the  temperature  of  the  mercury, 
and  that  of  the  air  at  the  two  stations,  and  to  substitute  these 
observed    elements    in    this    formula. 

Much  labor  is,  however,  saved  by  the  use  of  a  table  for  the 
computation  of  these  results,  and  we  now  proceed  to  explain  how  it 
may  be  formed  and  used. 


328  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Make 

60345,51  [1   +  (t4  +  t'  -64°)0,00102]  =  A, 


,  =  B. 


Then  will 


1  —  0,002551  cos  2-1 

1 

1  +  (T  -  T)  0,0001 


=  C. 


z  —  A  B  >  log 


h 
z  =  AB>  [log  C  +  log  ht  —  log  h]  ; 

and    taking   the    logarithms  of  both    members, 

log  3  =  log  A  +  log  B  +  log  [log  C  +  log  ht  —  log  A]  .  .  (404) 

Making  tt  +  t'  to  vary  from  40°  to  162°,  which  will  be  sufficient 
for  all  practical  purposes,  the  logarithms  of  the  corresponding  values 
of  A  are  entered  in  a  column,  under  the  head  A,  opposite  the 
values  tt  +  t',  as   an    argument. 

Causing  the  latitude  -^  to  vary  from  0°  to  90°,  the  logarithms 
of  the  corresponding  values  of  B  are  entered  in  a  column  headed 
B,  opposite   the   values   of  4'« 

The  value  of  T  —  T'  being  made,  in  like  manner,  to  vary 
from  —  30°  to  +  30°,  the  logarithms  of  the  corresponding  values 
of  C  are  entered  under  the  head  of  C\  and  opposite  the  values  of 
T  —  T'.  In  this  way  a  table  is  easily  constructed.  Table  IV  was 
computed  by  Samuel  Howlet,  Esq.,  from  the  formula  of  Mr.  Francis 
Baily,  which  is  very  nearly  the  same  as  that  just  described,  there 
being   but   a   trifling  difference    in    the    co-efficients. 

Taking  Equation  (464)  in  connection  with  Table  IV,  we  have  this 
rule  for   finding  the   altitude  of  one    station   above   another,  viz.  : — 

Take  the  logarithm  of  the  barometric  reading  at  the  lower  station, 
to  which  add  the  number  in  the  column  headed  C,  opposite  the  ob- 
served value  of  T  —  T ',  and  subtract  from  this  sum  the  logarithm 
of  the  barometric  reading  at  the  upper  station ;  take  the  logarithm 
of  this  difference,  to  which  add  the  numbers  in  the  columns  headed 
A  and  B,  corresponding  to  the  observed  values  of  tt  +  t'  and  -^  ; 
the  siim  will  be    the    logarithm    of  the   height    in    English  feet. 


>  MECHANICS    OF    FLUIDS  329 

Example. — At   the   mountain  of  Guana  xuato,  in  Mexico,  Von  Hum 
boldt  observed   at   the 

Upper  Station.  flower  Station. 

Detached    thermometer,    t'  =  70c,4 ;      tt   =  77°,6 
Attached  "  T'  =  70,4 ;       T  =r  77,6. 

Barometric  column,  h  =  23,66  ;     ht  —  30,05. 

What  was    the  difference  of  level  1 
Here 

tg  +  f  =  148°  ;     T  —  T  =  7°,2  ;     Latitude  21°. 

in. 

To  log     30,05  =  1,4778445 
Add  C  for  7°,2  =  9,9996814 

1,4775259 

in. 

Sub.  log  23,66  =  1,3740147 

Log  of     -    -    -     0,1035112  =  -  1,0149873 

Add  A  for  148°  -    -    -    -     =        4,8193975 

Add  B  for  21°    -     -    -     -     =        0,0008689 

ft.  - 

6843.1 3,8352537; 

whence   the    mountain   is   6843,1  feet  high. 

It  will  be  remembered  that  the  final  Equation  (464)  was  deduced 
on  the  supposition  that  the  air  is  in  equilibrium,  that  is  to  say, 
when  there  is  no  wind.  The  barometer  can,  therefore,  only  be  used 
for  levelling  purposes  in  calm  weather.  Moreover,  to  insure  accu- 
racy, the  observations  at  the  two  stations  whose  difference  of  level 
is  to  be  found,  should  be  made  simultaneously,  else  the  temperature 
of  the  air  may  change  during  the  interval  between  them  j  but  with 
a  single  instrument  this  is  impracticable,  and  we  proceed  thus,  viz. : 
Take  the  barometric  column,  the  reading  of  the  attached  and  detached 
thermometers,  and  time  of  day  at  one  of  the  stations,  say  the 
lower;  then  proceed  to  the  upper  station,  and  take  the  same 
elements  there )  and  at  an  equal  interval  of  time  afterward,  observe 
these  elements  at  the  lower  station  again ;  reduce  the  mercurial 
columns  at  the  lower  station  to  the  same  temperature  by  Equation 
(394),  take  a  mean  of  these  columns,  and  a  mean  of  the  tempera- 
tures   of    the    air   at   this   station,  and   use   these   means  as   a   single 


330  ELEMENTS     OF     ANALYTICAL     MECHANICS 

set   of  observations    made    simultaneously    with    those    at    the   highei 
station. 

Example. — The    following   observations    were    made    to    determine 
the   height   of  a  hill   near  West  Point,  N.  Y. 

Upper  Station.  Lower  Station. 

(1)  (2) 

Detached   thermometer,  V  =  57°  ;       tt   =  56°      and  61°. 
Attached  "  T  =  57,5  ;     T  =  56,5     and  63. 

in.  in.  in. 

Barometric  column,  h   =  28,94 ;  ht  =  29,62  and  29,63. 

First,    to    reduce   29,63    inches   at    63°,    to    what    it   would    have 
been    at  56°, 5.     For    this   purpose,  Equation    (394)  gives 


in. 


k  (I  +  T  -  T  X  0,0001)  =  29,63  (1  -  6,5  x  0,0001)  =  29,611 
Then 

\  =  °±+«l  -  -  =  58o,5,  ■ 

lt  -f  t'  =  5S°,5  +  57°-  -  =  115°,5, 
T—  T  =  56°,5  -  57°,5-  =  -  1°. 

in. 

To  log  29,6155   =  1,4715191 
Add  C  for  —  1°  =  0,0000434 

1,4715625 

Sub.  log  of  28,94  =  1,4614985 


Log  of  -  -  -  -  0,0100640  =  -  2,0027706 
Add  A  for  115°,5  -  -  -  =  4,8048112 
Add  B  for  41a,4    -     -     -     -     -        0,0001465 


/* 


642,28 2,8077283; 

whence  the  height   of  the   hill  is  642,28  English  feet. 


MOTION    OF    HEAVY    INCOMPRESSIBLE    FLUIDS    IN    VESSEia. 

§  282. — Let  it  now  be  the  question  to  investigate  the  flow  of  a  heavy, 
brmogeneous  and  incompressible  fluid  through  an  opening  in  a  vessel 
which  contains  it.     And    for  this  purpose,  resume  Eq.  (407),  which  is 


MECHANICS    OF    FLUIDS. 


331 


directly  applicable  to  the  case.     The  only  incessant  force  being  the  weight 
of  the  fluid,  take  the  axis  z  vertical  and  positive  upwards;   then  will 

X  =  0 ;      Y  =  0  ;     and     Z  =  —  g. 

Any  lateral  or  horizontal  motions  will  have  no  vertical  components 
and  may,  therefore,  be  disregarded,  and  we  shall  have,  Eq.   (405), 

which  will  reduce  the  general  Eq.  (407)  to 


and  integrating, 


&» 


d(p 
p^-Dgz-D-jj 


4  D .  «*  4    C 


•    (465) 


Next,   find  the   function   <p,   and  its  differential  co-efficient  of  the  first 
order  with  respect  to    t. 
Equations  (405)  give 

d  <p  =  to  .  d  z  , 
<p  =/*  •  d  z (466) 

Let  A  B  C  D  be  a  vertical  section  of  tie  vessel,  and  take  the  following 
notation,  viz. : 

*    =  the  variable  area  of  the  stratum 

whose  velocity  is  to. 
§   =  the  constant  area  of   any  deter- 
minate horizontal  section  of  the 

vessel,  as  C  D. 
S  =  the  area   of  the  section   of   the 

vessel  by  the  upper  surface  of  the  I ~. 

liquid;   this  may  be  constant  or 

variable,  according  as  the  upper 

surface  is  stationarv  or  movable. 


Z 


332  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

wt  ==   velocity  of  the  stratum  parsing  the  section  s{  at  C  Z>,  at  the  time  t 


The  continuity  of  the  fluid  requires  that 


w  •  s  =  Wj  •  st  , 


because  the  same  quantity  must  flow  through  every  horizontal  section  it 
the  same  time  :  whence 


w  =  w.  — : 


which,  in  Eq.  (466),  gives 

the  integration  being  taken  with  respect  to  the  variable  2,  of  which  s  is  a 
function.     This  function  will  be  given  by  the  figure  of  the  vessel,  h  being 
the  height  of  the  upper  surface  of  the  fluid  above  the  opening. 
It  mav  be  well  to  remark  here,  that 


z 

r 

J  z  +  h 


9      dz 
* 


will  be  constant  for  the  same  vessel  and  same  value  of  h;  and  if  the 
figure  of  the  vessel  be  that  of  revolution  about  a  vertical  axis,  it  will  only 
be  necessary  to  have  the  equation  of  this  vertical  section  to  find  the  value 
of  the  integral.     The  quantity  h  is  called  the  head  of  fluid. 

Differentiating  <p  with  respect  to  /,  and  recollecting  that  the   velocity 
downward  is  negative,  we  find 

d  <p  d  10       •<     d  z 

dt  d t  J  z+h    s    ' 

and  this,  with  the  value  of  v,  in  Eq.  (465),  gives 

d  w    fd  z         ^  w'1     .v,2         _       ..  n„. 
,  =  -  08t  +  D   .     ~j~  -  D-£  ■  $  +  C  ■  (407) 


MECHANICS    OF    FLUIDS.  333 

To  find  the  value  of  C,  let  p  =  P  t ,   when   z  =  z4 ,  which   corre. 
sponds  to   the  section   CD  of  the   liquid  ;  then  will 

^  ^  «*      dw.      f      dz        _   w.2      s *        „ 

Pl=-D91ll+D.Sl.lr±.jt=--D-t.i}+C, 

which,    subtracted  from   the  equation   above,  gives 

«  -r^r  v        TN         dw.  Pzdz        ^w^Vs,2  T    , 

,,_^  =  -iMs-0+i>-v^//T-^|>- xJ-(468) 

Also,  if  P'  denote  the   pressure   at   the   upper   surface    corresponding 
to  which  z  =  z\  we  have 

Now  z'  —  z,  ■=.  h  =  height  of  the   fluid  surface  above   the  section 
CD)   whence,  by  substitution  and   transposition, 

The   quantity  of  fluid   flowing   through  every  section  in.  the  same 
time   being   equal,  we   also  have 

—  Sdk  =  st  ,wt.dt.     •     *     •-••■•     *    •     (4T1); 

By  means  of  this  equation,  t  may   be   eliminated  from    Equation' 

(470) ;    then   knowing  the  quantity  of  the  liquid,  the  size  and  figure 

yn%t  d  z        r^dz 
—  ==  /    — ♦ 

in  which  s  is   a   function  of  z. 


*t    * 


rl  in 

§  283. — The  value  of    —r— -   being  found  from  Equation  (470),  and 

(X   C 

substituted  in  Equation  (468),  this  latter  equation  will  give  the  value 
of  the  pressure  p  at  any  point  of  the  fluid  mass  as  soon  as^w,  be- 
comes  known. 

Two  cases  may  arise.  Either  the  vessel  may  be  kept  constantly 
full  while  the  liquid  is  flowing  out  at  the  bottom,  or  it  may  be 
suffered   to   empty  itself. 

§  284. — To  discuss    the  case  in  which   the  vessel  is  always  full,  or 
the   fluid   retains   the  same  level  by  being  supplied  at  the  top  as  fast 


334        ELEMENTS    OF    ANALYTICAL    MECHANICS. 

as   it   flows    out   at   the    bottom,    the   head   h    must   be    constant,  and 
Equation  (471)  will  not  be  used. 
Making,  in  Equation   (4V0), 

A  =  2tJ.T> 


£  =  2g(h  +  IL*L); 


I>9 


s? 


n  _  zi 1  • 

and   solving  with  respect   to  d  t,  we   have 

■  «  =  £&? V  <472> 

Now,  three  cases  may  occur. 

1  st.  S  may  be  less   than  s, ,  and  C  will  be  positive. 

2d.    S  may  be  equal   to  sn  in  which   case  C  will   be  zero. 
,     3d.    S  may  be  greater  than  s, ,  when  C  will  be  negative,  and  this 
is  usually  the  case  in  practice. 

In  the  first  case,  when  C  is  positive,  we  have,  by  integrating  Equa- 
tion (472),  and  supposing  t  =  0,  when  wt  =  0, 

t  =  —==  •  tan     wt  K/rg  ; (473) 

whence, 

W/=v/|.tan^.t     .    .    .    .    .     (474) 

from  which  we  see  that  the  velocity  of  egress  increases  rapidly  with 
*he  time;   it   becomes   infinite  when 


a >  '  t  ~:  IT 


w 


t  = -==-     •    • (475) 

When  (7=0,  then  will  the   integration  of  Equation  (472)  give 


A 
~B 


***•**> •     '     '     <4™) 


MECHANICS    OF    FLUIDS.  335 

or  replacing  A  and  B  by  their  values,  and  finding  the  value  of  wt , 


"<  = =QT '; <4r» 


whence,  the  velocity  varies  directly  as  the  time,  as  it  should,  since 
the  whole  fluid  mass  would  fall  like  a  solid  body  under  the  action 
of  its  own  weight. 

When   C  is   negative,  the   integration  gives 


A           .       VB  +  w.  *JC 
t  =  ,  •  log  -*—= * ; 

2-y/lfC  ^B-Wty/U 


whence, 


•  t 

"<  =  '*U*7l'\/i!1 (478) 

e    A        +1 

in  Which  e  is  the  base  of  the  Naperian  system  of  logarithms  =  2,718282. 
If  the  section    S  exceeds   st  considerably,  the   exponent    of  e  will 
soon    become   very  great,  and  unity  may  be  neglected  in  comparison 
with    the   corresponding    power  of  e  ;  whence, 


w-  =  \/J  =  \/ : t^ — ;•  •  •  («») 


that  is  to   say,  the  velocity  will    soon  become  constant. 

If  the  pressure  at  the  upper  surface  be  equal  to  that  at  the  place 
of  egress,  which  would  be  sensibly  the  case  in  the  atmosphere, 
P'  —  Pt  =  0,  and 


(480) 


and  if  the  opening  below  become  a  mere  orifice,  the  fraction 


s2 


—  =  0; 

and 

u>t  =  V27A; (481) 


336        ELEMENTS    OF    ANALYTICAL    MECHANICS. 

that  is  to  say,  the  velocity  with  which  a  heavy  liquid  will  issue  from 
a  small  orifice  in  the  bottom  of  a  vessel,  when  subjected  to  the  pressure 
of  the  superincumbent  mass,  is  equal  to  that  acquired  by  a  heavy  body 
in  falling  through  a  height  equal  to  the  depth  of  the  orifice  below 
the  upper  surface  of  the  liquid,  which  is  called  the  law  of  Torricelli, 
who  discovered  it  experimentally. 

The  velocities  given  by  Equations  (479),  (480),  (481),  are  inde- 
pendent of  the  figure  of  the  vessel. 

If  the  velocity  wt  be  multiplied  by  the  area  st  of  the  orifice,  the 
product  will  be  the  volume  of  fluid  discharged  in  the  unit  of  time. 
This  is  called  the  expense.  The  expense  multiplied  by  the  time  of  flow 
will  give  the  whole  volume  discharged. 

§  285. — The  velocity  w4  being  constant  in  the  case  referred  to  in 
Equation  (479),  we   shall  have 

dt  * 

fcnd  Equation  (468)  becomes 

f  =  P,  -  Dg  (,  -  o  -  D.  \    (5l  _  i)  ; 

or,   substituting  the  value  of  w4 ,  given   by  Equation  (470), 

£<!*_  i 
p  =  Pt-Dg{z-  z,)  +  {D9h  +  P>  -  Pt)  .£_  ;    .     .     (482) 

whence,  it  appears,  that  when  the  flow  has  become  uniform,  the  pres 

sure   upon   any  stratum   is  wholly  independent   of  the   figure   of   the 

vessel,  and  depends  only  upon  the  area  *  of  the  stratum,  its  distance 

s  2 
from  the  upper  surface  of  the  fluid,  and  upon  the  ratio -^. 

§  286. — If  the  vessel  be  not  replenished,  but  be  allowed  to  empty 
itself,  h  will  be  variable,  as  will  also  S  except  in  the  particular 
cases  of  the  prism  and  cylinder. 

Making 

w,  =  y/^gff,  (483) 


MECHANICS    OF    FLUIDS.  337 

in  which  H  denotes  the  height  due  to  the  velocity  of  discharge :  we 
have 

dwt  =  *—=. (484) 


and,  Equation  (471), 


,                   S  •  d  h  .  ^ 

dt= * (485) 


and  by  integration. 


1             rS>dh 
t=  C =•    I —;= (486) 


To  effect  the  integration,  S  and  H  must  be  found  in  terms  of  A. 
The  relation  between  S  and  h  will  be  given  by  the  figure  of  the 
vessel.  Then  to  find  the  relation  between  H  and  h,  eliminate  wt  \ 
d  w4 ,  and  d  t  from  Equation  (470),  by  the  values  above,  and  we  have 

or,  dividing  by 

s*      rh  dz 

~s'Jo  V 

*  ■  (^-'  +  g  '•('-£ 

^-= dh  +  dE i-f B    dh  =0.(487) 

0   /*  dz  n   Ph  dz  v        ' 

'Jog  'Jos 

and  making 


'Job  '    J  o       8 


Qdh  +  dlf  +  RJIdk  =  0.       .     .     .  (488) 

fRdh 

Multiplying  by  e        , 

fRdk  f^dh  fRdh 

dh>Qe         +dH>e         +-  H  .  e         xRdh=0; 


338  ELEMENTS    OF    ANALYTICAL    MECHANICS. 


or 


J  Rdh  ,        ftldh. 

dh-  Q-e         +  d  (He         )  =  0; 

and  integrating 

//Rdh  /Rdh 

dh-Q-e        +  He         =  C\  .    .    .    .     (489) 

whence, 

—/Rdh     .  /Rdh. 

H=  e        •    (C  -    fdh-Q-e        )'   '     ' 


(490) 


The  constant  must  result  from  the  condition,  that  when  H  =  0, 
h.  must  be  ht,  the  initial  height  of  the  fluid  in  the  vessel. 

Thus  H  becomes  known  in  terms  of  A,  and  its  value  substituted 
in  Equation  (486)  will  make  known  the  time  required  for  the  fluid 
to  reach  any  altitude  h.  The  constant  in  Equation  (486)  must  be 
determined,  so  that  when   t  =  0,  h  =  ht. 

§  28*7. — Equation  (490)  gives  a  direct  relation  between  S,  A,  and 
H\  the  figure  and  dimensions  of  the  vessel  give  another  between  S  and 
h.  From  these,  two  of  the  three  variables  mav  be  eliminated  from  the 
Equation  (486)  and  the  integration  performed.  Take,  for  example,  the 
case  of  a  right  cylinder  or  prism.  Here  S  will  be  constant,  and  equal 
to  s. 


L 


h     ilz        h 

7  ~~s' 


Moreover,   let    us    suppose    P'  —  Pt  =  0,  which    would    be    sensibly 
true  were  the  fluid    to   flow  into   the    atmosphere    that    surrounds    the 

vessel.     Also,  for  the  sake  of  abbreviation,  make  —  =  &,  ther   will 

K 

and 

J  Rdh  =  (1  -  &)fj  =  (1  -*8)1ogfc 


MECHANICS    OF    FLUIDS.  339 


and  Eq.  (490)  becomes 

-*»)logA 

• 

[C-JWdh-e 

-*s)let 

1 

Multiplying  the  last  term  by 

2 

-k2    h 

2 

i 

-k2    K 

we  may  write 

-(l-ka)log  k 

H=e 

.[»- 

-.2 -**/rf('" 

(l-k»>  log  A 

— (1— l2)log  h 

=  e 

.[,. 

*2        ,      •»- 

-it*)  log 

1. 

when  H  •=.  0,  then  will  /i  =  A/ 

and 

C=2 

*2 

-  k2 

(1-42)  lor  ft, 

hre                   ; 

which  substituted  above, 

gives. 

after  reduction, 

*  =  2 

**.* 

-F 

-]' 

but, 

1 

ll-Jfcs 

e 

)losr     /AA1-*2; 

>] 


and  therefore, 

-^[(ir-]-^D-o"n---« 

which    substituted   in   Equation  (486),  gives 

Ik2  —  2      P  dh  ltMK 

in  which   the   only  variable  is  h. 

§  288. — The  particular  case  in  which  k2  =  2,  gives  to  this  value 
for  t  the  form  of  indetermination.  When  this  occurs,  we  must  have 
recourse  to  the  form  assumed  by  Equation  (488),  which,  under  this 
supposition,  becomes 

2kdh  +  hdff-  Hdh  =  0: 


340         ELEMENTS    OF    ANALYTICAL    MECHANICS 

—  2 

multiplying  by  h     , 

2h~ldk  +  h~l.dH  -H.hT2dh  =  0, 
2  • h  d  —  =  0, 

h  v .    * 

21ogA  +  y  =  (7; 

and  because  IT  =  0  when  h  =  h, , 

21ogA,  =  C; 
whence, 

#  =  2  A  •  log  y> 

and  this,  in  Equation  (486),  gives 

'  1       /»  G?A 


vT* 


\/2A-Io«T 


Making  ~  =  — ri   this  becomes  between  ine  limits  .r  =  0,  a:  =  1 


X' 


t  =  C 


§  289. — If  the   orifice  be   very  small   in   comparison   with   a  cross 
section  of  the  prismatic  or  cylindrical   vessel,  then  will  H  =  A,  and 

Equation  (486)  gives 

< 

<=<7--^L--/I 
Making   t  =  0   when  h  =  J),  we  have 

« =  -M=  .  (y>T,  _ -/*).  •     •     •     •     •     (494) 
and  for  the  time  required  for  the  vessel  to  empty  itself,  h  =  0,  and 


MECHANICS    OF    FLUIDS.  311 

Now,  with   the   same   relation  of  the   orifice    to    the   cross   section 
of  the  cylindrical  vessel,  we   have,  Equation  (481), 

wt  —  y/2yh, 

and   for  the    volume    of    fluid   discharged   in   the    time   t,    when   the 
vessel   is    kept  full, 

wt  .  st.t  =  s4,t .  -\Z2gh, 

and    if  this   be   equal    to  the   contents    of  the   vessel, 

*t.t.  y/2ghl  =  S .  ht ; 
whence, 

S         fh. 


s4      V  2g 

That  is,  Equation  (495),  the  time  required  for  a  prismatic  or  cylin- 
drical vessel  to  discharge  itself  through  a  small  orifice  at  the 
bottom  is  double  that  required  to  discharge  an  equal  volume,  if 
the  vessel  were   kept  full. 

§  290. — The    orifice    being    still    small,    we    obtain,    from    Equa- 
tion (485), 

d  h  §  , 

whence  it  appears  that,  for  a  cylindrical  or  prismatic  vessel,  the 
motion  of  the  upper  surface  of  the  fluid  is  uniformly  retarded.  It 
will  be  easy  to  cause  S  so  to  vary,  in  other  words,  to  give  the 
vessel  such  figure  as  to  cause  the  motion  of  the  upper  surface  to 
follow  any  law.  If,  for  example,  it  were  required  to  give  such  figure 
as  to  cause  the  motion  of  the  upper  surface  to  be  uniform,  then 
would  the  first  member  of  the  above  equation  be  constant  ;  and, 
denoting   the  rate  of  motion   by  a,  we   should   have 


whence, 

s?  .2g  h 


&  = 


a* 


but  supposing   the    horizontal    sections   circular, 

^  .  »?'2gh 

22  a* 


342      ELEMENTS    OF   ANALYTICAL   MECHANICS, 
and,  therefore, 


whence  the  radii  of  the  sections  must  vary  as  the  fourth  root  of  their 
distances  from  the  bottom.  These  considerations  apply  to  the  con- 
struction of  Clepsydras  or  Water  Clocks. 


STEADY     FLOW     OF     FLUIDS. 

§  291. — When  with  lapse  of  time  the  form,  quantity,  density, 
pressure  and  velocity  of  a  stream,  at  and  for  any  given  point  therein, 
do  not  change,  though  differing  from  one  point,  or  cross  section,  to 
another,  then  the  flow  is  steady  or  permanent;  and  under  such  cir- 
cumstances the  general  equations  for  fluid  motion,  (400)  and  (401), 
become  greatly  simplified. 

Moreover,  as  the  flow  of  water  in  rivers,  channels,  and  pipes  may 
generally  be  considered  steady  for  short  periods  of  time,  or  its  mean 
value  for  longer  durations,  very  many  of  the  most  important  appli- 
cations of  fluid  motion  are  only  problems  of  steady  flow. 

From  the  definition  given  it  is  clear  that  the  steady  motion  of 
fluids  will  be  expressed  by  Equations  (399)  and  (400),  if  the  five 
quantities,  u,  vy  w,  p,  *nd  D  are  assumed  not  to  be  functions  of 
time,  but  only  of  x,  y,  zy  the  co-ordinates  of  position ;  or,  analyti- 
cally, if 

du       dv       dw       dp       dD 

It  ~~  di  ""  ~dt  ~  It  ~  ~di  ~~     * 


which  conditions  give 


d?x       d*y       d?z 

Jv2    1a  "* 


dt*  '  '  di2  '     dt* 
and  cause  Equations  (399)  to  become, 


whence  also, 


dx  dy  dz 


dU  =  Xdx  -f  Ydy  -f  Zdz  =  —  dp. 


MECHANICS    OF    FLUIDS.  343 

These  are  the  same  as  Equations  (417)  and  (423);  which  proves 
that,  in  the  uniform  or  steady  flow  of  fluids,  pressure  acts  as  it  does 
when  they  are  in  a  state  of  equilibrium  of  rest. 

Multiplying   now  the  first   of   Equations  (399)    by  e/.r,  the  second 
by  dy,  and  the  third  by  dz,  then  adding,  we  obtain, 

1    ,  ,„        ,   Jdx*       dy*       d*\ 

3d*=dn^*d\dfi+M+m 

But  the  quantity  within  brackets  is  the  square  of  the  velocity ;  and 
we  have,  therefore,  for  the  general  law  of  steady  motion  of  any  fluid, 

lll  +  vdv  —  dll  =  0.    .     .     .     .     .     ,f    (496) 

For   gases   obeying   Mariotte's   law  (§  244),  if   the   temperature  is 
constant, 

D-P-- 
■        P' 

and  substituting,  we  have,  by  integration, 

p  \ogp  _+_ _  _  n  =  a (496') 

Also,  for  liquids,  D  being  constant, 

|  +  J-n  =  C. (497) 

If  now  the  motion  of  the  liquid  be  due  only  to  its  weight,  then 

dU.  =  Zdz  =  —  gdz, 


and 

D 

or  integrating,  dividing  by  g,  and  reducing, 


J?  4-  vdv  +  gdz  =  0 ; (497') 


z  +  2l  +  -  =  H, (497") 

T  to  T  2g  '  v         ' 

a  very  important  formula,  which  is  called  the  theorem  of  Bernouilli. 


344       ELEMENTS    OF    ANALYTICAL   MECHANICS. 

§  292. — To  interpret  which,  let  z  and  z'  be  the  vertical  ordinate*, 
or  heights  above  the  co- 
ordinate horizontal  plane 
xy,  for  the  successive  posi- 
tions of  the  same  particle 
in  a  steadily  flowing 
stream.  Then  it  has  been 
shown  in  §  (266)  that,  if 
<o  denote  the  specific  grav- 
ity of  the  liquid, 


0                  0' 

n 

m 

■^           ""**"*»« 

n 

Z 

\ 

m' 

Z' 

a 


a' 


P 

—  =  mn 


will  be  the  height  of  the  liquid  column  which  exerts  the   pressure  p 
upon  the  unit  of  surface.     Also,  the  laws  of  falling  bodies  give 


v* 


on  =s 


2/ 


for  the  height  due  the  velocity.  Hence  we  have  for  the  dynamic 
head,  or  total  height  of  fall, 

ao  =  H  =  z  -\-  mn  -f  no  =  z'  -f  m'n'  +  n'o' ; 
or, 

which,  therefore,  shows  that  the  fall  from  the  level  plane  oo'  to  the 
co-ordinate  plane  xy,  or  the  dynamic  head,  is  constant;  but  that  it 
divides  itself  into  three  portions,  one  due  to  the  velocity  v,  another 
due  to  the  pressure  p,  and  the  third  being  the  residual  height,  or  the 
co-ordinate  z,  of  the  particle  above  the  co-ordinate  plane  xy,  or  that 
of  lowest  level. 


§  293. — The   theorem  of   Bernouilli   is   readily  deduced  from  our 
fundamental  law, 

•  cPs 

1  Pdp  —  2m  —  ds  as  0, 
dtz 


MECHANICS    OF    FLUIDS 


345 


a 

^  "#a\ 

^---^y  \  \ 

^^^       V           N. 

^^           \          ^. 

^*                     X                 ^k 

\^ 

x            x 

\             X 

*         \ 

\        \ 

\         \ 

x               v     ^K^\ 

w.\ 

Z 

z' 

\        *        \ 

\        *        \ 

\        »        \ 
\        * 

\        \ 

\        * 

when  integrated  between  limits  and  written  in  the  form 

2  f  Pdp  =  ±-2m(v*  —  v*). 
Jo 

For  this  purpose  we  need  only  find  the  work  done,  and  its  equiv- 
alent change  of  kinetic  energy.  Let 
the  flow  be  steady,  and  ab  be  a 
portion  of  the  stream  which  sub- 
sequently occupies  the  equal  volume 
ab'.  Also,  let  z  and  z'  be  the  co- 
ordinate heights  of  a  and  6,  the 
centres  of  the  cross  sections,  whose 
areas  are  «  and  «',  and  whose 
mean  velocities  are  v  and  v '.  Then 
as  the  volume  q,  which  flows  in 
through  the  area  a  in  the  unit  of 
time,  must  be  the  same  as  that  which  passes  out  through  a',  we  must 

have 

q  =  av  =  av'. 

And  the  work  done  positively  by  the  pressure  p  at  a,  and  negatively 
by  p    at  6,  will  be 

(p—p')q; 
to  which,  if  we  add 

«(*  —  *')&       » 
the  work  in  the  fall  of  the  weight  o>q  from  z  to  z',  we  get 

w(^-0?  +  (p-p')q, 

the  total  amount  of   work  done  by  the  pressure   and   by  tke  weight, 
which  are  the  only  forces  supposed  to  work. 

For  the  equivalent  change  of  living  force,  we  have 


(D 


-  („*  _  V2)  q  ; 


and,  therefore, 


346      ELEMENTS    OF    ANALYTICAL    MECHANICS. 

or  transposing, 

p        v2         ,       p        v'2 

which    is    Equation    (498),    or   the   theorem    of   Bernouilli    as    fouud ' 
before. 

§  294. — By  comparing  this  theorem  with  its  analogous  formula, 

found  in  §  266,  for  the  head  of  a  heavy  liquid  at  rest,  it  appears 
that  the  only  difference  of  head,  in  passing  from  problems  of  rest  to 
those  of  steady  motion,  is  the  addition  of  the  term 

v2 

— -  =  071, 

or  the  height  due  to  the  velocity  of  flow.  And  conversely,  if  the 
velocity  v  be  zero,  then  there  is  no  motion,  and  the  theorem  of  Ber- 
nouilli, Equation  (497"),  reduces  to  Formula  (435"),  that  of  the  total 
head  of  a  heavy  liquid  at  rest. 

§  295. — Multiplying  Equation  (497")  by  2>mg,  the  weight  of  the 
mass  2  m,  we  obtain, 

2 


(p\  V* 

z  -f  — )  +  2  m  —  ==  2  mg  H, 


an  expression  stating  the  total  potential  energy,  or  stored  work,  due 
to  the  elevation  H,  to  be  equal  always  to  the  sum  of  the  variable 
potential  energy 

s*f  (*+!), 

added  to  its  supplementary  kinetic  energy,  or  to  the  half  sum  of  its 
living  force. 

And  this  is,  evidently,  only  the  reverse  of  the  demonstration  by 
which  Equation  (498)  has  just  been  obtained  from  our  general  fun- 
damental law. 


MECHANICS    OF    FLUIDS.  347 

§  296. — If  we  suppose  a  jet  to  spout  from  a  small  orifice  a,  in 
the  bottom  or  side  of  a  vessel,  kept  constantly  filled  to  the  same  level 
z,  where  the  sectional  area  a  is  so  large  that  its  velocity 

i 
a    , 
v  =  — v 
a 

can  be  neglected;   and  if  the  atmospheric  pressure  be 

then  Equation  (498)  becomes, 

j£=(«-V)=A'.     .......     (499) 

and  we  have, 

v'  =  \/2ffh/, (499') 

the  law  of  Torrecelli  for  the  velocity  of  a  spouting  jet. 

§  297. — It  should  be  borne  in  mind  that  these  laws  are  deduced 
under  the  hypothesis  that  there  is  no  friction,  viscosity,  or  other 
obstruction  preventing  particles  from  gliding  without  loss  of  velocity 
upon  each  other  and  upon  enveloping  surfaces.  This  is  far  from 
being  true ;  and,  consequently,  to  calculate  their  real  flow  corrections 
must  be  applied.  We  shall  give  but  one  example,  interesting  histor- 
ically, and  in  its  uses  important,  that  of  the  contracted  vein. 

A  stream  flowing  through  an  orifice  is  called  a  vein.  In  estimating 
the  quantity  of  fluid  discharged,  it  is  supposed  that  there  are  neither 
within  nor  without  the  vessel  any  causes  to  obstruct  the  free  and  con- 
tinuous flow ;  that  the  fluid  has  no  viscosity,  and  does  not  adhere  to 
the  sides  of  the  vessel  and  orifice ;  that  the  particles  of  the  fluid  reach 
the  upper  surface  with  a  common  velocity,  and  also  leave  the  orifice 
with  equal  and  parallel  velocities.  None  of  these  conditions  are  ful- 
filled in  practice,  and  the  theoretical  discharge  must,  therefore,  differ 
from  the  actual.  Experience  teaches  that  the  former  always  exceeds 
the  latter.  If  we  take  water,  for  example,  which  is  far  the  most 
important  of  the  liquids  in  a  practical  point  of  view,  we  find  it  to  a 
certain  degree  viscous,  and  exhibiting  a  tendency  to  adhere  to  surfaces 


348       ELEMENTS    OF    ANALYTICAL   MECHANICS. 

with  which  it  may  be  brought  in  contact.  When  water  flows  through 
an  opening,  the  adhesion  of  its  particles  to  the  surface  will  check 
their  motion,  and  the  viscosity  of  the  fluid  will  transmit  this  effect 
towards  the  interior  of  the  vein ;  the  velocity  will,  therefore,  be 
greatest  at  the  axis  of  the  latter,  and  least  on  and  near  its  surface ; 
the  inner  particles  thus  flowing  away  from  those  without,  the  vein 
will  increase  in  length  and  diminish  in  thickness,  till,  at  a  certain 
distance  from  the  orifice,  the  velocity  becomes  the  same  throughout 
the  same  cross  section,  which  usually  takes  place  at  a  short  distance 
from  the  aperture.  This  effect  will  be  increased  by  the  crowding  of 
the  particles,  arising  from  the  convergence  of  the  paths  along  which 
they  approach  the  aperture ;  every  particle  which  enters  near  the  edge 
tending  to  pass  obliquely  across  to  the  opposite  side.  This  diminution 
of  the  fluid  vein  is  called  the  veinal  contraction.  The  quantity  of  fluid 
discharged  must  depend  upon  the  degree  of  veinal  contraction  and 
the  velocity  of  the  particles  at  the  section  of  greatest  diminution ; 
and  any  cause  that  will  diminish  the  viscosity  and  cohesion,  and 
draw  the  particles  in  the  direction  of  the  axis  of  the  vein  as  they 
enter  the  aperture,  will  increase  the  discharge. 

Lagrange  gives  the  following  account  (Mec.  Anal.,  2e  partie,  sect.  X) 
of  the  discovery  of  the  contracted  vein :  "  Newton  tried  to  demonstrate 
the  law  of  Torrecelli  in  the  second  book  of  his  Principia,  which 
appeared  in  1684;  but  that  passage  is,  it  must  be  avowed,  the  least 
satisfactory  of  any  in  his  great  book.  Having  measured  the  flow  of 
water  from  an  orifice,  during  a  given  time,  he  thence  concluded,  in 
the  first  edition,  that  its  velocity  of  escape  is  only  that  due  to  half 
the  height.  This  error  arose  from  his  not  having  then  observed  the 
contraction;  but  in  the  second  edition,  which  appeared  in  1714,  he 
corrected  it,  by  stating  that  the  smallest  section  of  the  vein  is  to  the 

orifice  nearly  as  1  to  V2;    so  that,  taking   this   section    for  the   true 

area,  the  velocity  must  be  increased  in  the  ratio  of  y2  to  1,  and 
thus  it  becomes  that  due  to  the  height,  as  found  by  Torrecelli." 

§  298. — For  steady  flow  from  a  vessel  kept  filled  to  a  constant 
height,  the  contraction  is  calculable  in  the  particular  case  of  discharge 
through  an  adjutage  of  Borda,  which  is   only  a  re-entrant  cylindrical 


MECHANICS    OF    FLUIDS 


349 


tube,  so  short  that  the  liquid  escapes  without  touching  its  inner  .sur- 
face. Such  a  tube  causes  the  pressure  within  the  vessel  to  distribute 
itself  as  in  case  of  equilibrium :  so  that  the  pressure  on  the  area  a 
of  its  orifice  will  always  be  the  same  and  equal  to  that  on  s,  its  pro- 
jection on  the  opposite  side  of  the  vessel.. 

Omitting  the  atmospheric 
pressure,  which  acts  equally 
and  oppositely,  the  pressure 
on  the  area  «  of  the  orifice, 
or  mouth  of  the  tube,  will  be 

p  a  •=.  0)  h!  a, 

which  produces  during  the 
lapse  of  time  dt,  the  quan- 
tity of  impulsion, 

padt  =  G)k'  adt. 

For  the  contracted  vein  a!  the  equivalent  impulsion  is  that  of   the 
mass  escaping  in  the  same  time  dt,  multiplied  by  its  velocity,  or 


1 

•' 

a 

s 

• 

(G)    .    .  ■    \         .       a)  a  v  2  _ 
-a'v'dt)  x  *'  = dt. 


And  these  equal  values  give 


a)  a  h!  — 


oa  v 
9 


but,  by  the  law  of  Torrecelli, 


and,  therefore, 


v'2  =  2ghr, 
2a'  =  a; 


or  the  contracted  vein  has  an  area  equal  only  to  0.5  of  that  of  the 
tube.  The  velocity  through  the  orifice  a  will,  therefore,  be  only  one- 
half  of  that  through  the  contracted  neck,  or 

v  =  0.5  ^2gh'. 


Experience    shows   that   the   greatc  st   contraction  takes  place ,  at   a 


350        ELEMENTS    OF    ANALYTICAL    MECHANICS. 


distance  from  tho  vessel  varying  from  a  half  to  once  the  greatest 
dimension  of  the  aperture,  and  that  the  amount  of  contraction  depends 
somewhat  upon  the  shape  of  the  vessel  about  the  orifice  and  the 
head  of  fluid.  It  is  further  found  by  experiment  that  if  a  tube  of 
the  same  shape  and  size  as,  the  vein,  from  the  side  of  the  vessel  to 
the  place  of  greatest  contraction,  be  inserted  into  the  aperture,  the 
actual  discharge  of  fluid  may  be  accurately  computed,  provided  the 
smaller  base  of  the  tube  be  substituted  for  the  area  of  the  aperture; 
and  that,  generally,  without  the  use  of  the  tube,  the  actual  may  be 
deduced  from  the  theoretical  discharge  by  simply  multiplying  the 
theoretical  discharge  into  a  coefficient  whose  numerical  value  depends 
upon  the  size  of  the  aperture  and  head  of  the  fluid.     So  tha. 


a v  =  C a  \2gJi, 

in  which  C  is  the  coefficient  and  a  the  sectional  area  of  the  orifice. 
Moreover,  all  other  circumstances  being  the  same,  it  is  ascertained 
that  this  coefficient  remains  constant,  whether  the  aperture  be  circular, 
square,  or  oblong,  which  embrace  all  cases  of  practice,  provided  that 
in  comparing  rectangular  with  circular  orifices,  we  compare  the  small- 
est dimension  of  the  former  with  the  diameter  of  the  latter.  The 
value  of  this  coefficient  depends,  therefore,  when  6ther  circumstances 
are  the  same,  upon  the  smallest  dimension  of  the  rectangular  orifice, 
and  upon  the  diameter  of  the  circle,  in  the  case  of  circular  orifices. 
But  should  other  circumstances,  such  as  the  head  of  fluid,  and  the 
place  of  the  orifice,  in  respect  to  the  sides 
and  bottom  of  the  vessel,  vary,  then  will 
the  coefficient  also  vary.  When  the  flow 
takes  place  through  thin  plates,  or  through 
orifices  whose  lips  are  bevelled  externally, 
the  coefficient  corresponding  to  given  heads 
and  orifices  may  be  found  in  Table  V, 
provided  the  orifices  be  remote  from  the 
lateral  faces  of  the  vessel.  This  table  is 
deduced  from  the  experiments  of  Captain 
Lesbros,  of  the  French  engineers,  and  agrees 
with  the  previous  experiments  of  Bossut,  Michelotti,  and  others. 


MECHANICS    OF    FLUIDS. 


51 


As  the  orifice  approaches  one  of  the  lateral  faces  of  the  reservoir, 
the  contraction  on  that  side  becomes  less  and  less,  and  will  ultimately 
become  nothing,  and  the  coefficient  will  be  greater  than  those  of  the 
table.  If  the  orifice  be  near  two  of  these  faces,  the  contraction 
becomes  nothing  on  two  sides,  and  the  co- 
efficient will  be  still  greater. 

Under  these  circumstances,  we  have  the 
following  rules : — Denote  by  C  the  tabular, 
and  by  C  the  true  coefficient  corresponding 
to  a  given  aperture  and  head ;  then,  if  the 
contraction  be  nothing  on  one  side,  will 

C  =  1,03  C; 

if  nothing  on  two  sides, 

C"  =  1,06C; 
if  nothing  on  three  sides, 

C"  =  l,12  C; 

and  it  must  be  borne  in  mind  that  these 
results  and  those  of  the  table  are  applicable 
only  when  the  fluid  issues  through  holes  in 
thin  plates,  or  through  apertures  so  bevelled 
externally  that  the  particles  may  not  be 
drawn  aside  by  molecular  action  along  their 
tubular  contour. 

When  the  discharge  is  through  thick  plates  without  bevel,  or 
through  cylindrical  tubes  whose  lengths  are  from  two  to  three  times 
the  smaller  dimension  of  the  orifice,  the  expense  is  increased,  the  mean 
coefficient,  in  such  cases,  augmenting,  according  to  experiment,  to 
about  0,81.5  for  orifices  of  which  the  smaller  dimension  varies  from 
0,33  to  0,66  of  a  foot,  under  heads  which  give  a  coefficient  0,619 
in  the  case  of  thin  plates.  The  cause  of  this  increase  is  obvious. 
It  is  within  the  observation  of  every  one  that  water  will  wet  most 
surfaces  not  highly  polished  or  covered  with  an  unctuous  coating — 
in  other  words,  that  there  exists  between  the  particles  of  the  fluid 
and  those  of  solids  an  adhesion  which  will  cause  ihe  former  to  spread 
themselves    over   the    latter   and    stick    with    considerable    pertinacity. 


352       ELEMENTS    OF    ANALYTICAL    MECHANICS. 

This  adhesion  becoming  effective  between  the  inner  surface  of  the 
tube  and  those  particles  of  the  fluid  which  enter  the  orifice  near  its 
edge,  the  latter  will  not  only  be  drawn  aside  from  their  converging 
directions,  but  will  take  with  them,  by  the  force  of  viscosity,  other 
particles  with  which  they  are  in  sensible  contact.  The  fluid  filaments 
leading  through  the  tube  will,  therefore,  be  more  nearly  parallel  than 
in  the  case  of  orifices  through  thin  plates,  the  contraction  of  the  vein 
will  be  less,  and  the  discharge  consequently  greater. 


STEADY     MOTION     OF     ELASTIC     FLUIDS. 

§  299. — As  in  the  case  of  incompressible,  so  also  in  that  of 
elastic  fluids,  if  we  suppose  the  motion  to  have  been  established  and 
become  permanent,  the  velocity  of  a  stratum  as  it  passes  any  partic- 
ular cross  section  of  the  vessel  will  always  be  constant,  and  the  quan- 
tity of  fluid  which  flows  through  every  cross  section  will  be  the  same ; 
while  its  density  and  volume  may  vary  from  one  position  of  the 
section  to  another.  All  lateral  velocity  will  be  disregarded.  And  the 
motion  will  be  supposed  to  be  due  only  to  the  weight  of  the  ele- 
ments and  to  the  elastic  force  arising  from  some  external  force  of 
compression. 

Our  fundamental  formula,  ■   ' 

cPs 
2iPdp  —  2?n—ds  =  0, 

o-ives 

2  P  cos  a  —  2  m  —  =  0 (a) 

at 

Therefore,  we  need  only  find   the   forces   and   their  equivalent   accel- 
eration. 

Let  z  and  z  —  dz  be  the  vertical  co-ordinates  of  the  same  particle 
hi  the  two  consecutive  cross  sections  a  and  a\  at  the  distance  ds 
from  each  other.     Then,  the  same  mass, 

—  ads  =  —avdL 
9  9 

flows  in  the  same  time  through  each  of  them. 


MECHANICS    OF    FLUIDS.  353 

The  normal   component  of  its  weight  is 

—  .  a)  a  ds  =  <>)  a  dz, 
as 

which  is  negative  because  it  acts  downwards  and  z  is  taken  positive 
upwards. 

The   positive   pressure    on    a  is  p  a,  and  the  negative  pressure  or 
resistance  on  a'  is 

p  a  -}-  d  (p  a)  ; 

their  difference, 

—  a  dp  —  p  da, 

acting  in  the  direction  ds,  reduces  to 

—  a  dp; 

for  pda,  being  lateral,  has  no  effective  component  in  the  direction 
ds,  and  tends  only  to  increase,  or  diminish,  the  area  of  the  cross 
section  without  affecting  the  quantity  of  the  flow. 

Substituting  these  values  in  Equation  («),  we  obtain, 


o)adz  +  a  dp  -f-  i—av  dt\  —  =  0 ; 


or  reducing, 

.        dp       vdv  ♦     /,*„#» 

dz  +  J-^ —  0; (49V) 

(D  g  v 

and  from  this,  if  we  suppose  o)  constant,  integration  gives, 

P        v2 

which  is,  evidently,  the  same  as  Equation  (497"),  or  the  theorem  of 
Bernouilli. 

If  we  eliminate  the  constant  H,  we  get 


v 2  —  v«2 


L  =  <*— >  +  g-3 <498> 


2-7 
If  v0  be  very  small,  and  the  velocity  due  to  weight  from  z0  to  z  can 


354       ELEMENTS    OF    ANALYTICAL    MECHANICS. 

be  neglected,  which   is   always  the   case  when  the   pressure  p  is  the 
principal  moving  force,  then 

£=(2-9.-  ••••••  m 

a  formula  applicable  both  to  airs  and  liquids.     And  which,  if  we  put 
h  in  place  of  the  second  member,  gives 

v*  =  2gh,    ........     (499) 

or  the  law  of  Torrecelli. 

But  as  g>  is  not  constant  in  the  flow  of  air,  integration  in  the 
manner  supposed  cannot  be  performed.  Seeking  its  value,  therefore, 
from  the  law  of  Mariotte  and  Gay  Lussac,  Equation  (390),  we  have 


and,  consequently, 


P 

p  =  A  a)  =  —  (1  +  a  6)  0), 

if 


vdv        .  .dp  , 

\-dz  +  A—  =  0 (501) 

g  p  v       ' 


If  in  this  the  temperature  0  be  supposed  coustant,  integration  will 


Bfive 


v2 


-f  z  +  .4  log;?  =  const, (501') 

which  is  the  theorem  of  Navier. 
It  may  be  put  under  the  form, 

?^=  (*„_*)+ ,4  log  J.    ....     (501") 

And  if  in  this  equation  we  make  v  equal  to  v0i  or  the  velocity  be 
uniform  for  different  cross  sections,  as  it  would  be  in  steady  flow 
through  a  long  and  smooth  cylindrical  pipe,  then 

z  —  z0  =  A  log  — , 
P 

which  is  the  same  as  Equation  (436)  for  the  equilibrium  of  airs  at 
rest. 


MECHANICS    OF    FLUIDS.  355 

As   the    expense,  or   quantity,  is   constant   in   steady  flow  for  any 
two  sections  a  and  «',  we  have, 

apv  =  ccoPoVq, 
and  this  gives 

Generally  the  compressing  force  p0cc0  greatly  exceeds  the  resistance 
pa,  and  we  may,  therefore,  make 

g=(2o-z)  +  ^logj;      ....     (501'") 

and  if  the  effect  of  weight  from  z0  to  z  can  be  neglected,  this  sim- 
plifies and  becomes 

—  =  A\oA (501w) 

2g  P 

If  p   and  p0  do    not   differ  much,  then   integration  is   practicable 
without  logarithms,  for  we  may  put 

and  we  shall  have 

Af*dp_     /"°dp  _/Po  —  Pi\. 

which  gives 

^=fc->+fe-S);  .  .  .  («., 

And  this,  when  v0  and  (z0  —  z)  can  be  neglected,  reduces  to 

V  =  2^-5^1), .     (500") 

an  approximative  formula,  much  used  and  the  same  as  Equation  (500). 
The  identity  of  these  equations  is  due  to  the  fact  that  in  one  th» 
specific  gravity  w  being  supposed  constant,  while  in  the  other,  though 
considered  variable,  it  is  replaced  by  its  mean  value,  assumed  to  be 
constant,    these    two    hypotheses    do    thus    substantially    become    one. 


356      ELEMENTS   OF    ANALYTICAL    MECHANICS. 

Moreover,  the  same  formula  was  first  obtained  for  liquids  by  Ber- 
nouilli;  and  it  is  clear  that  the  hypothesis  of  constant  density  must 
give  always  the  same  equations,  both  for  liquids  and  gases. 

The  preceding  equations  and  discussion,  due  to  Navier,  are  those 
given  in  most  of  the  treatises  on  Mechanics.  But  they  are  very- 
defective,  for  the  reason  that  the  density  of  air  in  its  flow  cannot  be 
supposed  either  to  remain  constant  or  to  vary  according  to  the  law  of 
Gay  Lussac,  Equation  (390),  as  assumed  by  Navier.  The  expansion, 
in  fact,  generally  takes  place  according  to  Equation  (391),  or  the 
law  of  Laplace. 

The  mechanical  theory  of  heat  has  furnished  more  exact  equations 
for  the  flow  of  airs,  but,  before  deducing  them,  some  facts  to  which 
reference  is  needed  should  be  given. 


DIGRESSION     ON     THE     ACTION     OF     HEAT     UPON     AIRS. 

i 

§  300. — The  work  of  expansion  done  by  any  substance,  as  by 
steam  on  a  piston,  is  expressed  by  the  integral  of  the  pressure  p  on 
the  unit  of  surface,  multiplied  by  the  area  a,  and  by  the  distance 
moved,  ds\   this  gives  for  it, 


/  pads  =    I  p dv, 


in  which  dv  is  the  elementary  variation  of  volume. 

The   total  work,  or  energy,  of   any  system   may  be   expressed   by 
the  formula, 


JPdp  = 


U+S, 


in  which   U  is  internal  and  S  external  work.     If   S  denote  work  of 
expansion,  this  becomes 


^Jpdp  =  U  +  Jpdv. 


When  air  flows  by  its  expansion  into  a  partially  exhausted  re- 
ceiver, out  of  another  into  which  it  has  been  previously  compressed, 
the  mean  temperature    of   the  whole    system    does    not  vary.     Hence, 


• 


MECHANICS    OF    FLUIDS.  357 

if   the  internal  work    U  be  regarded    as   a   function    of  the  volume  v 
and  temperature  0,  its  variation  for  6  is  independent  of  v,  or 


<M§+§-£)- 


reduces  to 


dU=^dO; («) 


or  U  is  a  function  of  0  only.     Moreover,  observation  proves  that  for 
airs 

-jx  =  Ec  =  const.,  .......     (a) 

in  which    the    constant   tt    is    the    specific   heat   of    constant   density, 
(§  245),  and  i?  is  an  abstract  number. 
Integrating  this  result,  we  get 

u,-u,  =  Ec(dx-eQ) (j8) 

The  specific  gravity  of  any  substance  being  denoted  by  w,  its 
reciprocal  v  is  called  the  specific  volume,  and  their  product  is  the  unit 
of  weight.  The  law  of'  Gay  Lussac,  Equation  (390),  may  therefore 
be  expressed  by  the  formula, 

pv  =  ap0  v0  (-  +  Of  ; (y) 

or,  changing  the  origin  of  9,  by  making 

r  =  -  +  e (t) 

it  may  be  put  under  the  simple  form, 

pv  =  Rt, ,     •     .     (e) 

in  which  R  is  constant. 

For  differences  of  temperature, 

dd  =  dr; 

and 

(6  -  0O)  =  (r  —  t0). 


358       ELEMENTS    OF    ANALYTICAL    MECHANICS. 

The  new  origin  of  temperatures  is  thus  transferred  273°  centigrade 
below  the  melting  point  of  ice,  as  is  readily  seen  by  making  r  zero. 
And  r  is  called  the  absolute  temperature,  because  Equation  (e)  shows 
that  for  airs  the  elastic  force  of  expansion  p  becomes  nothing  when 
r  is  zero.  In  the  theory  of  heat  this  is  also  proved  to  be  true  for 
all  substances. 

Let  c  and  ct  be,  respectively,  the  specific  heat  of  constant  pressure 
and  constant  density ;  then  to  raise  the  temperature  of  the  unit  of 
weight  through  0  degrees  will  require,  if  the  pressure  be  kept  con- 
stant while  the  air  expands,  the  quantity  of  heat  cB ;  and  if  the 
volume,  or  density,  be  kept  constant,  only  the  quantity  cQ  will  be 
required;  the  difference  of  these  quantities  is  the  heat  which  does 
the  work  of  expansion;    and  we  shall,  therefore,  have 

E(c  —  c)=zap0v{i  =  R,     .• (£) 

The  constant  E  is  called  the  mechanical  equivalent  of  heat, 
because,  as  this  equation  shows,  when  units  of  heat  are  multiplied 
by  it,  they  are  transformed  into  the  expression  for  their  equivalent 
amount  of  mechanical  work ;  and  this  algebraic  transformation  merely 
expresses  the  physical  reality.  So  that  E  is  an  abstract  number, 
which  for  French  measures  has  been  found,  experimentally,  to  be 
equal  to  425,  the  value  generally  adopted. 

But  this  ratio  E  may  be  calculated  from  the  equation  just  given. 
For  this  purpose  observation  has  given  for  air  the  following  data : 

c  =  0.2375,  c  =0.1684,  «  =  0.00367  ; 

the  barometric  pressure  0.760  m.  is  on  the  square  metre  10340  kilo- 
grammes ;  and  at  0°  centigrade  a  cubic  metre  of  air  under  0.760  m. 
pressure  is  found  to  weigh  1293  grammes.     Hence, 

_  ap0v0  _         10340  x  0.00367         _ 
=  c  _C/  "  :  1.293  (0.2375  —  0.1684)  '  ' 

a  value  agreeing  very  nearly  with  425,  that  given  by  the  most  exact 
experiments. 


MECHANICS    OF    FLUIDS. 


359 


NEW     EQUATIONS     OF     STEADY     FLOW. 

g  301. — Let  the  quantity  of  fluid  entering  with  steady  flow  through 
the  cross  section  a  be  the  unit  of  weight ;  the  same  quantity  escapes 
in  the  same  time  through  another  section  a  ;  let  also  u0  and  ux  be 
their  respective  velocities,  then 


1  /    Pdp  =  -± 
v  n 


u02 


o  2<7 

is  the  total  work  of  all  the  forces  both  external  and  internal. 

The  work  of  compression 
done  by  p0  upon  the  sur- 
face, or  section  a,  will  for 
the  unit  of  weight  be 


/  pads  ==  /   p dv  =  p0 «v 

And  for  the  escape  of  an 
equal  quantity  at  a  the  re- 
sistance will  be 


/. 


px  dv  =  px  vx. 


w>»»f>»w>»»»„jj*s»js,w;s;ss;;;,- 


a 


ra 


V 


a' 


N 


For  the  work  of   gravity  on    the   unit   of   weight   from    z0  to   z„  the 
heights  of  the  centres  of  gravity  of  a  and  a\  we  have  simply 

(z0  —  zx)  =  h. 

And  denoting  the  internal  energy  by  U,  we  have,  for  the  total  work 
of  all  the  forces, 


Uj2  —  Mq8 

*9 


=  (*o  -  *i)  +  (Po^o  —Pi  »,)  +  (*7o  —  CTm), . .  .  (502) 


a  general  equation  for  all  fluids,  whether  liquid    or  gaseous;    but   for 
which  we  may  find  a  simpler  form. 

The  internal  work  is  only  that  of  expansion  from  the  volume  v9 
under  the  pressure  p0  to  the  volume  vx  under  the  pressure  px.  It 
takes  place  for  a   particle  along   a  line  of   flow  mo ;    and   if   rapidly, 


360        ELEMENTS    OF    ANALYTICAL    MECHANICS. 

then  sufficient  time  is  not  allowed  for  heat  to  be  either  absorbed    or 
emitted.     We  have,  therefore, 


dv 


or  integrating  by  parts, 

piVl  —  p0v0  +    /    °vdp; 

which,  by  substitution,  gives 

ut2  —  u 


*9 


2  /JO 

°-  =  (z0  —  z1)+J    vdp (503) 


Three  distinct  cases  now  present  themselves,  and  we  shall  discuss 
them  successively  : 

Case  I. — The  density  of   the   fluid,  liquid  or  gaseous,  is  constant. 
Its  internal  work  will  be  zero,  and  our  general  Equation  (502)  reduces  to 

-      u2  +  z  jl.  P  -  -  U*  4.  z   +  P«  (±a«\ 

2^  +  *  +  u-27~Mo+u'      *     *     *     *      (498) 

or  the  theorem  of  Bernouilli.  Which,  if  u0  is  neglected,  as  is  always 
the  case  if  a  is  much  greater  than  a\  reduces  to  the  law  of 
Torricelli, 

u*=2g(z0-z), (499) 

when  external  pressure  is  neglected;  and  becomes  when  weight  is 
neglected, 

u2      Po—Pi 


2g  '         a 


(500) 


a  true  formula  for  liquids;  but  for  gases  merely  approximative,  and 
to  be  used  only  when  p0  differs  little  from  jo„  and  when  the  mean 
value  of  a>  may  be  assumed  to  be  constant. 

For  airs  there  must  be  loss  of  temperature,  for  the  equation 

pv  =  Rt 

gives,  if  v  be  supposed  to  remain  constant, 

v  dp  =  R  dr ; 


MECHANICS    OF    FLUIDS.  361 

• 

and  therefore, 

R 
Po—Pi  =  —  (~o  —  t,)  ; 

or  the  temperature  varies  proportionally  to  the  pressure.  This  is  sim- 
ply verified  by  feeling  the  breath  blown  upon  the  hand  to  be  warm 
or  cold  proportionally  to  the  amount  of  force  employed. 

Case  II. — The  fluid  is  supposed  to  be  an  air  obeying  the  laws  of 
Mariotte  and  Gay  Lussac,  but  receiving  heat  during  expansion,  so  as 
to  keep  its  temperature  constant. 

Then  we  have 

p  v  =  p0  v0  =  A  =  const. ; 
and  this  gives 

p  dv  =  v  dp  =  A  — . 

P 

which,  by  substitution  and  integration,  transforms  Equation  (503)  into 

U"  ~Ul  =  (*.  -Zl)+A  log* 


2<?  "v         "    ■  °p 


or  the  theorem  of  Navier,  as  expressed  by  Equation  (501").  But 
flowing  air,  which  is  an  extremely  bad  conductor,  can  very  rarely 
receive  during  expansion  sufficient  heat  from  external  sources  to  pre- 
vent its  being  chilled  by  its  own  expansion;  and  if  this  condition  is 
not  fulfilled  the  formula  of  Navier  is  inapplicable. 

Case  III. — The  fluid  neither  receives  nor  parts  with  heat  during 
expansion,  and  is  supposed  to  be  an  air  obeying  the  laws  of  Mariotte 
and  Gay  Lussac. 

We  have  then  the  conditions, 

p0v0  =  Br0;  pxvxz=Rr^ 

R=zE(c  —  cx)', 

UQ-Ul  =  Ecl(rQ-rx). 

And  by  substitution  in  Equation  (502)  and  reduction,  these  give  for 
the  steady  flow  of  permanent  gases  the  law 


362       ELEMENTS    OF    ANALYTICAL    MECHANICS. 


M,2  —  Ut? 


l-—-"-=(zQ-zx)  +  Ec(r(i-Tl).    .     .     .      (504). 

Which,  if  (z0  —  ^i)  and  u0  be  neglected,  reduces  to 

u2 

j-  =  Ec(t0—t1) (504') 

Now  to  apply  this  formula  we  must   find  r}  when  t0  is  observed, 
for  which  Equation  (391)  gives 


?,=&)"=&- 


then 

but 

p0c<v0c  =  plc<v1c, 

Ih  vo  ro 

and  these  give 

\  C—Ci 

%=We< <505> 


from  which   rl  is   easily  calculated  when  r0,  p0  and  jo,  are   given    by 
observation. 

As  an  example,  let  gas  escape  from  a  vessel  at  30°  C.  and  a 
pressure  of  one  and  a  half  atmospheres  into  the  air,  and  let  u0  be 
neglected,  then  we  have 

g  =  9.81  m. ;  p0  s  1.5 ;  j»i  =  1  *, 

r0=:303o;  c  =  0.2375;  c,  =  0.1684; 

and  these  data  give  for  r,  the  value, 

r,  =  r0  (f)o-2909  =269°.16. 

So  that  the  temperature  will  descend  to  nearly  4°  below  the  freezing 
of  water. 

And  this  result  causes  Equation  (504')  to  become, 

—  =  Ec  (303°  —  269°.16)  : 

which  gives  for  the  velocity  of  escape, 

u  m  250  m.  nearly. 


MECHANICS    OF    FLUIDS.  363 

We  have,  therefore,  three  formulas  for  the  steady  flow  of  airs — one 
that  of  Bernoulli,  Equation  (500),  which  can  only  be  used  when  p0 
and  px  differ  but  slightly  ;  another  that  of  Navier,  rarely  applicable ; 
and  the  third,  Equation  (504),  which  is  given  by  the  thermoelastic 
properties  of  air,  and  which  should  generally  be  employed. 

Analogous  equations  have  been  determined  for  steam,  which  does 
not  obey  the  law  of  Mariotte,  but  for  those  we  must  refer  to  treatises 
on  heat. 


PART    III. 


MECHANICS  OF  MOLECULES. 


8  302. — The  more  general  circumstances  attending  the  action  of 
ft  rces  upon  bodies  of  sensible  magnitudes  have  been  discussed.  They 
constitute  the  subjects  of  Mechanics  of  Solids  and  of  Fluids.  Those 
which  result  from  the  action  of  forces  upon  the  elements  of  both  solids 
and  fluids  remain  to  be  considered.  They  form  the  subject  of  Me- 
chanics of  Molecules ;  which  comprehends  the  whole  theory  of  Electrics, 
Thermotics,  Acoustics,  and    Optics. 

It  is  assumed  that  all  bodies  are  built  up  of  elementary  mole- 
cules in  sensible,  though  not  in  actual,  contact;  that  the  relative  places 
of  equilibrium  of  these  molecules  are  determined  by  the  molecular  forces, 
and  that  the  intensities  of  these  forces  are  some  function  of  the  dis- 
tance between  the  acting  molecules.  A  displacement  of  a  single  mole- 
cule from  its  position  of  relative  restT  will  break  up  the  equilibrium  of 
the  surrounding  forces,  and  give  rise  to  a  general  and  progressive  dis- 
turbance throughout  the  body.  It  is  proposed  to  investigate  the  nature 
of  this  disturbance,  the  circumstances  of  its  progress,  and  the  conduct 
of  the  molecules  as  they  become  involved  in  it. 

PERIODICITY    OF    MOLECULAR    CONDITION". 

§  303. — Molecular  motions  cannot,  like  the  initial  disturbances  which 
produce  them,  be  arbitrary ;  but  must  fulfil  certain  conditions  imposed 
by  the    physical    connections  which   "nite  the   molecules   into  a  system 


366  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

These  motions  are,  so  to  speak,  constrained  by  this  connection.  Let 
the  conditions  of  constraint  be  expressed,  as  in  §  213,  Mech.  of 
Solid?,  by 

L  =  0  ;     If  =  0  ;     L"  =  0  ;   &c (506) 

Z,  L\  L",  &c,  being  functions  of  the    co-ordinates  of  the   molecules 

Denote  by 

X,  T,Z;  X,  T,  Z';   Ac, 

the  accelerations  impressed  upon  the  molecules  whose  masses  are  ft,  m', 

&c,  in  the  directions  of  the  axes.     Equation  (313)  will  obtain  for  each 

molecule.     There  will  be  as  many  equations  as  molecules,  and  by  addi« 

tion,  we  find,  by  inverting  the  terms, 

There  will  be  three  co-ordinates  for  each  molecule.  Denote  tin- 
number  of  molecules  by  i\  the  number  of  Equations  (506)  of  condition 
by  m ;  then  will  3  i  —  m  =  n,  be  the  number  of  co-ordinates  which, 
being  given,  will  reduce  the  number  of  unknown  co-ordinates  to  the 
number  of  equations.  These  unknown  co-ordinates  may,  hence,  be  found 
in  functions  of  the  known,  and  the  places  of  the  molecules  at  any  in- 
&tant  determined. 

Denote  the  m  co-ordinates  by  x  y  z,  x'  y'  z\  <fec,  and  the  n  co-ordi« 
nates  by  a  /3  y,  a'  j3'  y',  &c. :   then  we  may  write, 

x  =  <p,  (a  /J  y  a',  &c.)  =  px ; 

V  =  <Py  («  P  7  <*■',  &c.)  m  py  ; 
z  =  (pf  (a  j3  y  a',  <fec.)  =  pt ; 


x'=  <p.'(aj3ya',  <fec.)  =  pm>; 
<fcc.  =  &c.  s=  <fec. ; 


also, 


Y=ip,(apya',&c.)  =  P,; 

Z  =  ij),  («]3y  a',  &c.)  =  P. ; 

X'  =  V>v(a  j9  y  a',  &c  )  =  P..; 

&c.  =  <fec.  =  &c. ; 

in  which  <p„  pr  <p„  <fcc,  i/),,  1/,^  i/f,  &c,  denote  any  functions  of  th.    co- 


MECHANICS    OF    MOLECULES. 


367 


ordi nates  a  /3  y,  a'  <fcc,  which   result   from    the   conditions  of  Equations 
(506)  and  the  process  of  elimination. 
At  any  time  t,  suppose 

a  3  and  y  to  become  a  +  £    (3  -f  */,    y  +  £'» 
a'  jT  and  y'  to  become  a'  +  f,  j3 '  +  ?/',  y'-f  £' ; 


and  suppose  the  increments  £  77  £,  £'  77'  £',  <fec,  to  continue  so  small 
during  the  entire  motion  as  to  justify  the  omission  of  all  termr  into 
which  their  second  powers  and  products  enter ;    then  will 

,   dP*     y  ,    d  P*         .   dpx  "j 

*  =  P*  +  TT7  "  f  +  -J7S  '  V  +  -3T7  *  ?  +  &c-'   &c-» 


rf 


a 


dj3 


«/  =  />,+ 


*  =  />,+ 


dP*     e+dPy     w    , 
da      *^  tfj3      '^ 


a(  y 

of  y 
dp,     y 


£  +  &c,  &c, 


d  y 


%  -f-  &c,  &c, 


►  .    .  (508) 


■      ■      ■      • J 

aa  dp  ay 


>.    .  (509) 


Z=P,+ 


d  a 


f  + 


rfP. 


1  + 


rfP. 

rfy 


4*  +  &c,  <fec, 


X'=P„-f 


From  Equations  (508)  we  have 


0?  a 


rfj3 


c?y 


d2y  = 


dP,     a%*.     dP*     jnL    ,    rf^ 


a 


<*2£  + 


dtf 


rf*i?  + 


*,  =  *£.*£+**.*,,+ 


rf  a 


rf/3 


a?y 
dy 


<fcc. 


•  d2  £  -f-   &c.,  &c, 

•  a?2  £  -f-   «fec,  &C, 
<fcc 


r 


(510) 


368 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


also  from  the  same 


6  x  = 
6y  = 
6z  = 


d  a 

d  a 

dPz 

d  a 


<*£  + 
<*£  + 


dp* 
dp 

dPy 

d(3 

dP* 

dp 


8ri  -f- 
6r]  + 
6?)  + 


d  y 

dpy 
d  y 

dp, 

d  y 


6  £  +  &c, 
dZ+  &c. 


(511) 


The  Equation  (507)  contains  three  times  as  many  terms  as  there 
are  molecules,  each  term  consisting  of  a  variation  with  its  coefficient. 
Eliminate  from  this  equation  X,  Y,  Z,  X',  &c,  d'2  x,  </2y,  d2  z,  dlx\  <fcc, 
and  dx,  6  y,  6 z,  6 x\  &c,  by  means  of  Equations  (509),  (510),  and  (511); 
collect  the  coefficients  of  6  £,  6  rj,  6  £,  6  £',  &c. ;  the  number  of  terms 
will  reduce  to  n,  this  being  the  number  of  the  co-ordinates  a,  ft,  y,  a', 
<fcc.  These  variations  are  independent  of  one  another,  since  the  co- 
ordinates a,  j3,  y,  a',  &c,  are  so.  The  coefficients  of  these  variations 
must,  therefore,  be  separately  equal  to  zero.  Performing  the  operation, 
omitting  all  the  terms  containing  products  and  powers  of  £,  7/,  £,  £', 
&c,  higher  than  the  first,  there  will  result  n  equations  of  the  form, 

iB.^+lF.^+lF.^  +  lG^+^ff.rj  +  lK.^A^^SU) 

in  which  D,  E,  F,  G,  H,  &c,  are  functions  of  the  differential  co-efficients 
in  Equations  (509),  (510),  and  (511);  and  A  consists  of  a  series  of  terms 
each  composed  of  two  factors,  one  of  which  is  either  Pm1  Py,  PfJ  or 
some  other  P  with  subscript  co-ordinate  accented. 

If  a  P  y,  a',  &c,  give  the  places  of  rest  of  the  molecules,  then  will 
PM  =  Py  =  Pt  =  &c.  =  0,  and  Equations  (512)  become 

d%  £  d2n  d2£ 

These  equations  are  satisfied  by  making 

?=22.iV  .sin(*Vp-  /•),/ 
7)  =  R  .  i\Tf  .  sin  (t .  v^p  —  r), 
S^R.N^.  sin  {t.^p-  >), 

S     - "~  ... 


[ 


(514) 


MECHANICS    OF    MOLECULES.  369. 

:n  which  R  and  r  are  arbitrary  constants,  and  p,  N '  N ,  N ' ,  &c,  are 
constants  to  be  determined.  For,  after  two  differentiations,  regarding 
£,  77,  £,  &c,  and  t  variable,  we  have 

d2  £ 

—t  =  -  B.Nrsm(tVp  —  r)p, 

.  dl  =  ~  H.Nn.sm{tVi-r)p, 

d*?  - 

jj  =  -£.iV^sm(tVp-r)p, 

<fcc,  <fec. 

1 

which,  substituted  in  Equations  (513),  give,  after  dividing  out  the  com- 
mon factor  R  .  sin  {t  \p  —  r), 

{£D.Nf  +  ^E.N  +^FJST)p-^GN-^HN  -  ZKN  =  0.  (515) 
v  t  n  Vr  £  r,  ^  v 

1 

Now,  there  being  n  of  these  equations;  n  —  1  of  them  will  give  the 
values  of  N ,  N ,  N ',  <fcc,  in  terms  of  N  ;  and  these  being  substituted 
in  the  nth  equation,  must,  from  the  form  of  the  equations,  give  a  result- 
ing equation  having  JV  as  a  common  factor  and  of  the  nth  degree  in  p. 
The  common  factor  N  will  divide  out.  The  quantity  p  will  have  n 
values.  The  values  of  JV ,  N.  N ,.  <fcc,  will  be  rational  fractions  01 
the  (n  —  l)th  degree  in  p,  having  a  common  denominator,  and  each  multiplied 
by  N„  which  is,  as  yet,  arbitrary.  Make  JY  equal  to  the  common  de- 
nominator, and  JV ,  JV,  N ,  &c,  will  be  expressed  in  symmetrical  func- 
tions of  p,  of  the  (n  —  l)tb  degree.  Each  of  the  quantities  JV ,  iV,  iV, 
«fcc,  will  have  as  many  values  as  p ;  and  each  of  the  increments 
f>  ^i  £  !')  &c,  will  also  have  n  values,  each  set  of  which  will  satisfy 
Equations  (513). 

But  Equations  (513)  are  linear;  not  only,  therefore,  will  each  of 
the  values  of  £,  77,  £,  £',  <fec,  satisfy  them,  but  their  respective  sums  sub- 
stituted for  f,  77,  £,  £',  &c,  will  also  satisfy  them. 

Denoting  the  roots  of  the  nth  equation  in  p  by  p,  p,,  p2,  <fec,  and 
using  the  subscript  figures  to  designate  the  corresponding  values  for 
the  other  letters,  the  general  solution  of  Equations  (513)  will  be 


370 


ELEMENTS    OF    ANALYTICAL    MECHANICS 


f  =B.N t.  sin  (t.y/p— r)-\-B1N'i  .  sin  (t.y/pl^-ri)-\-BiN'f  .  sin  (t .  </ p2— r8)-j-&c. 
i  it  *a 

i=B.N  .s'm(t.<Jp— r)-\-B1.N  .&\n(t.y/p1  —  r1)-\-Bi.]V    .&'in(t.y/pi—ri)-]-&c. 
l,  i-R.IT  ,t&n(t.Jf—r)-t-£l.ire  .&'m(t.y/p  —rl)-\-B<1.N'  .sin(£. y/pi-r^)-\-&c. 


R,  R{,  <fcc,  and  r,  rh  &c,  arc  arbitrary  constants,  in  these  complete  in- 
tegrals. They  mast  be  found  in  terms  of  the  initial  values  of  £  i\,  £, 
tfcc,  and  their  differential  coefficients.  They  are  small,  because  the  ori- 
ginal disturbance  is  supposed  small. 

§  304. — If  all  the  quantities  R,  Rlt  i?2,  &c,  except  the  first,  vanish. 
Equations  (516)  lose  all  their  terms  in  the  second  members  except  the 
first,  and  Equations  (508)  become 


r-A+(fe^+^+S?-*i+-)*--('.V?-r^ 


•dP-   wr    ,   dP-   at    ,   d'P- 


M«l) 


§  305. — If  the  roots  p,  p„  p»  &c,  be  real,  the  different  terms  in  the 
values  of  1, 7/,  £,  &c,  as  given  in  Equations  (516),  will  disappear  period- 
ically, and  the  precise  times  of  disappearance  of  each  will  be  found  by 
making 


t  Vp  —  r  =  an;     t  Vp\  —  r,  =  a tt  ;     t  Vp%  —  r2  =  an\    &c. 


or, 


an  +  r  an  +  r, 


Vp 


Vpx 


t  = 


a  n  +  f% 


Vfr 


;    <fec, 


in  which  a  is  any  whole  number  whatever.     The  intervals  of  disappear* 
ance  will  be 


Tr-t-r       ir  +  rt.     7r  +  5;    &c# 


Vp  VpT  Vf>» 


MECHANICS     OF    MOLECULES.  371 

When  these  intervals  are  commensurable,  then  will  £,  77,  £,  &c,  resume 
the  values  they  had  at  some  previous  time,  the  molecules  will  return  to 
their  former  simultaneous  places,  the  movement  will  become  periodical, 
and  the  period  will  be  equal  to  the  least  common  multiple  of  the  above 
intervals.  This  phenomenon  of  periodical  returns  of  molecules  to  their 
initial  places,  is  called  the  periodicity  of  molecular  condition. 

§  306. — From  Equations  (516)  it  is  apparent  that  each  and  every 
individual  of  a  system  of  molecules  in  which  the  connection  is  such  as 
to  leave  n  of  their  co-ordinates  independent,  may,  when  slightly  dis- 
turbed from  rest  in  positions  of  stable  equilibrium,  assume  a  number  n 
of  oscillatory  movements,  and  that  all  or  any  number  of  these  may  take 
place  simultaneously.  And  conversely,  whatever  be  the  initial  derange- 
ment of  such  a  system,  the  resulting  motions  of  each  molecule  may  be 
resolved  into  n  or  less  than  n  simple  components  parallel  to  each  of 
any  three  rectangular  axes.  Here  we  have,  under  a  different  form,  the 
principle  of  the  coexistence  of  small  motions. 

§  307. — Again,  let  £„  77,,  £„  <fec,  be  the  values  of  £,  77,  £,  &c,  when 
the  system  is  in  motion  by  the  action  of  one  set  of  forces ;  £,,  %,  £* 
ike,  when  under  the  action  of  another  set,  and  so  on — the  initial  con- 
dition being  determined  for  each  set  of  movements — then,  Equation 
(516)  being  linear,  will  the  resultant  values  of  £,  77,  £,  &c,  be  given  by 

£=£  +  &  + g,+  &c, 
77  =  77,  +  77,  +  773  -f   &c, 

£=?,  +  <;  +  ?,+  Ac; 

and  here  we  also  have  again  the  superposition  of  small  motions.  That 
is,  each  molecule  may  take  up  simultaneously  the  motions  due  to  each 
disturbing  cause  acting  separately  and  alone. 

§  308. — Equations  (513)  may  also  be  satisfied  by  making 

t  Vp 


z= 

R, 

1 

e 

> 

n  = 

R. 

,N  . 
n 

t\p- 

e 
tVp- 

r 
•  r 

S  = 

:R 

w    r 

e 

1 

372  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

which  give 

d*£  tVp-r 


dt% 


Ti=P-*'K-e 


d*S  Ar       t.yfi-r 

J¥=p.B.tf.e 

and  these  substituted  in  Equations  (513),  give  Equations  (515),  wiUi 
the  exceptions  of  the  signs  of  the  terms  which  are  independent  of  p. 
But  with  this  solution  there  would  be  no  limit  to  the  increase  of 
£,  J],  £,  £',  <fcc,  which  is  contrary  to  the  conditions  that  the  disturbances 
are  to  continue  small.  In  fact,  this  last  solution  supposes  the  molecules 
to  be  moved  from  positions  of  unstable  equilibrium;  the  other,  which 
is  the  case  of  nature,  from  stable  equilibrium. 

WAVES. 

§  309. — It  thus  appears  that  every  molecule,  subjected  to  certain 
conditions  of  aggregation,  may,  when  disturbed  from  its  place  of  rela- 
tive rest,  describe,  under  the  action  of  surrounding  molecules,  a  closed 
orbit.  The  disturbed  molecule  being  acted  upon  by  its  neighbors,  will 
react  upon  the  latter,  and  cause  them,  in  turn,  to  take  up  their  appro- 
priate paths;  and  the  same  being  true  of  the  next  molecules  in  order 
of  distance,  the  disturbance  will  be  progressive  and  in  all  directions. 
That  is,  an  initial  disturbance  of  a  molecule  at  one  time  and  place, 
becomes  a  cause  of  disturbance  of  another  molecule  at  another  time 
and  place.  While,  therefore,  any  molecule  Ax  is  travelling  over  its 
orbit,  the  disturbance  is  being  propagated  on  all  sides,  and  at  the  in- 
stant the  former  completes  its  circuit,  the  latter  will  have  reached  a 
molecule  A^  in  the  distance,  which  will  then,  for  the  first  time,  begin 
to  move ;  and  the  molecules  Ax  and  A%  will,  thereafter,  always  be  at 
the  same  relative  distance  from  their  respective  starting  points.  In  the 
same  way,  a  molecule  A3J  still  further  in  the  distance,  will  begin  its 
first  circuit  when  A%  begins  its  second  and  Ax  its  third,  and  so  on. 


MECHANICS     OF    MOLECULES.  373 

Between  the  molecules  At  and  A.„  as  also  between  At  and  A3i  &c, 
molecules  will  be  found  at  all  distances  from  their  starting  points  and 
moving  in  all  directions,  consistently  with  the  dimensions  and  shapes  of 
their  respective  orbits.  The  term  phase  is  used  to  express  the.  condi- 
tion of  a  molecule  with  respect  to  its  displacement  and  the  direction 
of  its  motion. 

Molecules  are  said  to  be  in  similar  phases,  when  moving  in  parallel 
orbital  elements  and  in  the  same  direction ;  and  in  opposite  phases, 
when  moving  in  parallel  orbital  elements  and  in  opposite  directions. 

The  particular  form  of  aggregation  assumed  by  the  molecules  be- 
tween the  nearest  two  concentric  surfaces  in  which  the  same  phases 
simultaneous] v  exist  throughout,  is  called  a  wave. 

A  surface  which  contains  molecules  only  in  similar  phases,  is 
called  a  wave  front.  This  latter  term  is  generally,  though  not  al- 
ways, applied  to  the  surface  upon  which  the  molecules  are  just  begin- 
ning to  move.  The  velocity  of  a  wave  front  will  always  be  that  of 
disturbance  propagation.  A  wave  length  is  the  interval,  measured  in 
the  direction  of  wave  propagation,  between  two  consecutive  surfaces 
upon  which  the  molecules  have  similar  phases. 

WAVE    FUNCTION. 

§  310. — Denote  the  masses  of  the  molecules  by  m,  m',  <fec. ;  the  co- 
ordinates of 

m     by    x.  y,  z, 

m'      "     x  +  A  x,      y  -f-  A  y,      z  -f-  A  z, 
m"     "     x  +  Ax',     y  +  Ay',     z  +  Az', 
&c,  &c,  &c,  <fec, 

and  the  distance  between  any  two  molecules,  as  m  and  m\  by  r ;  then 
will 


r  =  VAx*  +  Ay*  +  As9 (518) 

*  _  . 

Let  f(r)  be  the  intensity  of  the  reciprocal  action  between  m  and  m  ; 
in  which  /  denotes  any  function  whatever.  This  reciprocal  action  will 
determine  the  elastic  force  of  the  body. 

23 


374 


ELEMENTS    OF    ANALYTICAL    MECHANICS, 


Before  the  system  is  disturbed,  there  will  be  no  inertia  developed, 
the  inertia  teims  in  Equations  (B-)  will  disappear,  and  we  shall  have 
for  the  action  on  anv  molecule  as  m, 

A  x 

*/(>•).- =  0, 

2/«'V  =°'  I (519) 

As 

V«  •  -  =  o. 

Now  suppose  the  system  slightly  disturbed,  and  denote  the  displacement 
at  the  time  t  in  the  direction  of  the  axes  x,  y,  z,  respectively,  of 

«•      by  S  i\,  £ 

m'      "  |  +  A£  ?PHM,  ?+Af, 

m"     "  ?+Af,  7/  +  A77',  £+A^, 
&c,            <fec,            &c,  &c. 

Then,  denoting  the  change  in  r  by  Ar,  Equation  (518)  becomes 

r  +  A  /•  =  •(A a?  +  A£)2+  (Ay  +  A?/)2  +  (As  +  A£)2    .    (520) 

and  by  the  principle  of  the  superposition    of  small    motions,  Equations 
(  A  )  give  for  the  action  on  m, 

d*Z       _,,     L.     .    Az  +  Af    1 
m  .  -5-3-  =  2/  (r  -f-  A  r) . 


cFtj 


m-Jj=*f(r  +  *r)-- 


Ar 


Ay-f  A?? 


r  +  A  r 


r-r«   f 


(521) 


But 


a  r  '      r-j-Ar 


1  1       Ar       Ar8 

— r~7~  = r*i — i —  &c« 

r  +  Ar       r        r  r 


/(r  +  Ar)=/(r)  +  ^Ar+k, 


whence,  neglecting  the  powers  of  A  r  higher  than  the  first, 


MECHANICS    OF    MOLECULES. 


375 


/  (r  +  A  r) 
r  +  £r 


•""•rffif^r? )"}•<"«» 


Squaring  Equation  (520),  neglecting  the  squares  of  A  r,  A  f,  A  17, 
and  A  £,  and  subtracting  the  square  of  Equation  (518)  from  the  result, 
we  find 

Aar.Af-fAy.Ajj  +  Az.A^ 


A  r  — 


.     .     .     .     (522) 


Substituting  this  above,  and  making 


Equations  (521)  become 

m  .  -j-J  =  2  j<p  (r) .  Af  -f  i/>  (r)  (Ax .  Af  +  Ay  .  At)  +  Az  .  A£)  .  Asj 

!»,-J J=S{q)(r).Aiy+V»(»")(Aaf.A|  +  Ay.Aiy  +  A2.A5).Ay| 

m.jji  =Z\y(r).Ar+i}>(?-)(Ax.AZ  +  A!/.A7)  +  Az.AZ).Azl 


(523) 


(.  (524) 


Pe>*forming  the  multiplication  as  indicated  in  the  last  term  of  the  sec- 
ond members,  there  will  result  terms  of  the  form, 

2  ij)  (r)  .  A  7)  .  A  x  .  A  y  ;       Zip  (r)  .A^.Az.Az;      lip  (r)  .  A  %  .  A  x  .  A  y  ; 

2^(r).A»|.Ay.Az;      2  i/>  (r)  .  A  £  .  A  z  .  A  y  ;      2i/>(r).Ag.A:r.Az; 

and  it  may  be  shown  by  the  process  of  §  163,  to  prove  the  existence 
of  principal  axes,  that  the  co-ordinate  axes  may  be  so  taken  as  to  cause 
these  terms  to  vanish.  Assuming  the  axes  to  sa'isfy  these  condition*, 
Equations  (524)  become 

d*l 


m  .  jj ?  =  2  { <p  (r)  +  y>  (r)  A  *»  j  A  & 


\. 


•       •      • 


(525) 


07/' 


ELEMENTS     OF     ANALYTICAL     MECHANICS, 


Making 


m  p'   =  <p  (r)  +  ip  (r)  .  A  x*A 

.Az\J 


mp"  =  9  (r)  +ip(r).  Ay\ 
mp'"=  <p  (r)  +  V*  (r) 


(526) 


Equations  (525)  take  the  form, 


d?~ 

2p' 

**i 

d>T] 
dt%    ~ 

2p 

.AT], 

dt*    ' 

2p'".AZ. 

(527) 


An  initial  and  arbitrary  displacement  of  a  molecule  at  on«  time 
and  place,  becomes,  through  a  series  of  actions  and  reactions  of 
the  molecular  forces  alone,  the  cause  of  displacement,  of  another 
molecule,  at  another  time  and  place.  In  this  latter  displacement, 
which  results  alone  from  the  molecular  forces,  the  molecular  motions 
must  take  place  in  the  direction  of  least  molecular  resistance.  This 
direction  is  at  right  angles  to  that  of  wave  propagation  ;  for,  the  force 
which  resists  the  approach  of  any  two  strata  of  molecules  will  be  much 
greater  than  that  which  opposes  their  sliding  the  one  by  the  other. 
Indeed,  this  view  is  abundantly7  confirmed  bv  manv  of  the  phenomena 
that  result  from  wave  transmission  ;  and  it  will  be  taken  for  granted, 
without  further  remark,  that  the  molecular  orbits  are  in  planes  at  right 
angles  to  the  direction  of  wave  propagation. 

§  311. — The  first  of  Equations  (527)  appertains,  therefore,  to  wave 
propagation  in  the  plane  y  z,  the  second  in  the  plane  x  z,  and  the  third 
in  the  plane  xy. 

The  integrations  of  Equations  (527)  are  given  by 

2  7T/Tr  v     1 


|  =  ae  .  sin  —  ( 

.      2  77  ,  __  x 

fj=  a,,  sin  -j-  (P,    t  —  rX 

Ay 

£=  a,,  sin  -rr  (Vt.t  —  rf), 


(528) 


MECHANICS    OF    MOLECULES. 


377 


in  which  Vx,  Vf,  and  Vt  are  the  velocities  with  which  the  disturbance 
is  propagated  in  directions  perpendicular  to  the  axes  a*,  y,  and  z,  re- 
spectively; Aa,  X9,  and  X,  the  shortest  distances,  in  the  same  directions, 
between  the  places  of  rest  of  anv  two  molecules  that  mav  have  at  the 
same  instant  the  same  phase;  r,,  rfJ  and  rz  the  distances  of  any  mole- 
cule's place  of  rest  from  that  of  primitive  disturbance,  estimated  in  the 
sime  direct!  /ns.     This  being  understood,  we  have  the  relations, 


rm  =t  ^y*  +  z* ;     r¥  =  v^  +  z2 ;     r2  -—  %V  _j-  y2. 


2tt  2tt  2tt 

t-  p,  =  ». ;    -3-  ^  =  »f ;     y-  ^  =  *. ;  r 

A,  A„  A, 


27T  27T  2tt 

Aj  A,y  A., 


and  the  above  become 


J  =  «. .  sin  (nx.t  —  hx  .  rx), 
77  =  ay  .  sin  (ra,  A  —  ky.  ry),  )■ 
£  =  a2 .  sin  (wz  .  *  —  &2  .  r2).  _ 


(529) 


(53(>) 


§  312. — To  show  that  these  are  the  solutions  of  Equations  (52*7),  it 
will  be  sufficient  to  prove  that  they  will  satisfy  those  equations  with 
real  values  for  nz,  ny,  and  nz.  Differentiate  twice  with  respect  tc  t, 
and  we  have 


d7 

de 


=  —  n*2  •  £ 
=  -  nj .  r\, 


(531) 


Give  to  rm,  rr,  and  rt  the  increments  ArJf  A  ry,  and  Ar1(  respectively; 
the  corresponding  increments  of  £,  7/,  and  £  are  A  |,  A  77,  and  A  £  and 
Equations  (530)  become 


378  ELEMENTS    OF    ANALYTICAL    MECHANICS. 


$  +  *$ 


am  .  sin  (nx  .  t  —  k,  .  rx  +  Jem  .  A  r.), 
ay .  sin  («, . *  —  ky  .  ry  -f  ky  .  A  rj, 
a,, .  sin  {nt .  t  —  kt .  i\  -\-  kt  .  A  rt). 


Develoj  ing  the  second  members,  regarding  nz  .  t  —  k,  .  r»,  ny .  t  —  ky .  rf 
and  nt,t  —  kt,rt  as  single  arcs;  subtracting  Equations  (530)  in  order 
replacing  1  —  cos  km  Ar„  1  —  cos  ky  A  rr,  and  1  —  cos  kz  A  rz  by  tlieii 
respective  values,  we  find 


*. .  rm), 


(kx  A  rm) 
A  %  =  —  2  £ .  sin2  ~ -  4*  sin  {K  &  rm)  •  am  cos  (n, .  t 

m 
h 

(£  A  r  ) 
A  tj  =  —  2  ?| .  sin2        -        +  sin  (&r  A  ry) .  ay  cos  («, .  t  —  £y  .  ry), 

(&  A  r ) 
A  £  =  —24".  sin2  — i— — -  +  sin  (&e  A  rj  .  «r  cos  (nz.t  —  kz  .  rE). 


►    (532) 


Substituting  these  in  the  second  members  of  Equations  (527),  we  have 


dp 

dP 


(A  A*'  ) 

-2g.Sp'.sm»     *      *  +     S/.sin(*BAr,).«  .cos(»a.<-*x.r#), 

-  2  if .  SJ9"  .  sin2  — ^ S-  -f-     %p"  .  sin  (£  A  rj  .  «   .  cos  (n   .  t  —  k  .  r  ), 

jj  "         ™  "  »  y       ™ 

=  -  2  { .  Z jpT .  sin3 -f-     £  /?"' .  sin  (*  A  rt)  .  «t .  cos  (»g .  *  -  *  .  rf). 


K588) 


Tn  the  state  of  equilibrium  of  the  molecules,  we  may  suppose 
their  masses  equal,  two  and  two,  and  symmetrically  disposed  on 
either  side  of  that  whose  mass  is  m.  Indeed,  this  is  the  most  general 
way  in  which  we  may  conceive  the  equilibrium  to  exist.  Then,  since 
for  every  positive  arc  km .  A  rx  there  will  be  an  pqual  negative  one,  we 
must  have 


2  p'  .  sin  (fct .  A  7\)  .  a.  .  cos  (ne  .  t  —  hm  .  ra)  as  0, 
2  p"  .  sin  {ky  .  A  ry)  .  ay  .  cos  (ny  .  t  —  ky  .  ry)  as  0, 
Sy ' .  sin  (kt .  A  r,) .  a, .  cos  (w, .  t  —  &, .  rt)  =»  0,  „ 


►  .     .     (534) 


and  therefore, 


MECHANICS    OF    MOLECULES. 


379 


dt2 
d¥ 

d¥ 


2  £  .  2  p    .  sirr , 

r  2 


2^2/\sin2^— *, 
2  £.2;?     .sin' , 


v    .     .     .     .     (535) 


whence,  Equations  (531)  and  (535), 


w,2  =  2  2//.  sin2  — -, 


n*  =  22/.  sin 


'//    _i_i 


n2  =r  2  2;/".  sin 


2        ' 


(536) 


which  are,  Equations  (526)  and  (522),  real  values  for  v„  »fJ  and  ft,. 

§  313. — Substituting  the  values  of  w,,  w^,  nz,  and  f^  &y,  &M  Equa- 
tions (529),  there  will  result,  after  multiplying  the  first,  second,  and 
third  by  1  =  A  r2  -f-  A  rj  ;  1  =  A  r2  -^  A  r2 ;  1  =  A  r2  -f-  A  r£  re- 
spectively, 


sinJ 


n  A  ra    1 


F.!  =  I2/   .Ar,2. 


/7r  A  rsy 


sin 


t  tt  A  ry 


F2 


^".Ar,2. 


r.f  =  j2y.Ar,2. 


.  2  7T  A  rf 
sin1*  — r — 

(7T.  Ar,v» 


>  .     .     .     .    (537) 


380 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


WAVE    SECTION. 


§314. — Resuming  either  of  Equations  (528),  say  the  first,  viz.: 


v  .       2  7T 

£  —  ax  sin  —  (  vx  .  t 

A_ 


it  is  apparent  that  if  t  be  made  constant  and  rx  variable,  so  as  to  reach 
in  succession  all  the  molecules  in  its  direction  between  the  limits 

Vm  .  t  —  X„    and    Vx  .  t, 

the  displacement  £  will  also  vary,  and  from  zero  to  zero,  passing 
between  these  limits'  through  the  maximum  values  aa  and  minimum 
value   —  ol,  ;    thus   deter- 


as  i 


mining  the  curved  'line 
C  D,  of  the  annexed 
figure,  to  be  the  locus 
of  the  corresponding  dis- 
placed molecules,  of  which  the  places  of  rest  are  on  the  straight  line 
A  B,  coincident  in  direction  with  the  line  rx  in  the  plane  y  z.  And 
it  is  also  apparent  that  if  the  above  value  of  t  receive  an  increment, 
making  the  time  equal  to  t\  and,  with  this  new  value  for  the  time,  rt 
be  made  to  vary  between  the  limits 

Va.t'  —  a.,    and  Vx.t', 

the  locus  of  the  corresponding  displaced  molecules  will  be  found  to 
have  shifted  its  place  to  C  D\  in  the  direction  towards  which  the  dis- 
turbance is  propagated. 

This  peculiar  arrangement  of  a  series  of  consecutive  molecules,  by 
which  the  latter  are  made  to  occupy  the  various  positions,  arranged  in 
the  order  of  continuity  about  their  places  of  rest,  is,  as  we  have  seen, 
§  305,  called  a  wave,  and  the  functions,  Equations  (528),  from  which  a 
section  of  the  waves  may  be  constructed,  are  called  wave  functions. 


WAVE    VELOCITY. 


§  315. — From    either    of   Equations   (537),  say  the    first,   it   appears 
that  the  velocity  of  wave  propagation  depcr  js  upon  the  ratio  between 


MECHANICS    OF    MOLECULES.  381 

7T  .  A  T 

the  arc  — -= — -   and   its  sine.     If  the  distance  A  r#,  between  the  mole- 

cules,  in  the  direction  of  r„,  have  any  appreciable  value  as  compared 
with  the  wave  length  A,,  this  ratio  will  be  less  than  unity;  and  in 
proportion  as  the  wave  length  increases,  in  the  same  medium,  will  the 
velocity  increase.  When  the  distance  A  r9  is  insignificant  in  compari- 
son with  the  wave  length  Ax,  the  ratio  of  the  sine  to  the  arc  will  be 
unity,  and  that  factor  will  cease  to  appear. 

§  316. — If  the  medium  be  homogeneous,  then  will 


p'  =  p"  =  p'"  ;     A  r„  =  A  ry  =  A  r, ; 

and,  therefore, 

V  =v  =v 

That  is,  the  velocity  will    be   the  same  in  all  directions. 

Denote   this 

velocity  by  V\   we  may  write 

sin                   .- 

Vs -IT.             k       

/7T  .  A  r\2 

in  which  the  two  factors  that  compose  the  second  member  have  such 
average  values  as  to  give  a  product  equal  to  the  sum  of  the  products 
which  make  up  the  second  members  of  either  of  Equations  (537). 

Supposing,  in  addition  to  the  existence  of  homogeneity,  that  the  in- 
terval between  the  molecules  is  insignificant  in  regard  to  the  wave 
length,  the  last  factor  of  Equations  (537)  reduces  to  unity,  and  taking 
the  axis  x  in  the  direction  of  the  velocity  to  be  estimated,  Ar  becomes 
Aar,  and,  first  of  Equations  (537), 

replacing  p'  by  its  value,  Equations  (526)  and  (523), 

2  m       L    r  Vrfrr*         r  /  J 

The  distances  between  the  molecules  being  very  small,  the  term  of 
which  A  a;4  is  a  factor  may  be  neglected  in  comparison  with  that  con- 
taining A  .t2,  and  the  above  may  be  written 


382  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

P*  =  -L.2/(r).—  .Ax. 

i  ... 

A  x 
.  Now, /(r). —   is  the  component  of  the  elastic  force  exerted  betweer 

two  molecules  whose  distance  is  r,  in  the  direction  of  the  axis  x;   and 

f(r). — -  .Ax    is    the    quantity    of    work    of    this    component    acting 

through  a  distance  Ax.     Making 

we  may,  by  the  principle  of  parallel  forces,  write 

*/ v)  •  ~~r • A x  —  2 e*  xi » 

in  which  e{  is  the  sum  of  the  component  molecular  forces  which  act  on 
one  side  of  the  molecule  ra,  in  the  direction  of  the  axis  #,  or,  which 
is  the  same  thing,  the  elastic  force  limited  to  a  single  molecule;  and 
rt  the  path  over  which  this  force  would  perform  an  amount  of  work 
equal  to  that  measured  by  the  first  member.     Substituting  this  above, 

F2  =  ^. 
m 

.Denote  by  i  the  number  of  molecules  in  a  unit  of  length,  and  multiply 
both  numerator  and  denominator  by  f\    we  have 

f  *2  * 

TT2   _  l    'e,'lX,  . 
—  •*  ? 

V .  m 

but  i2 .  et  is  the  elastic  force  extended  to  a  unit  of  surface,  and  is  the 
measure  of  the  elastic  force  of  the  medium;  call  this  e.  The  factor 
i  xt  is  the  number  of  molecules  in  the  distance  xt ;  call  this  k.  The 
denominator  &m  is  the  quantity  of  matter  in  a  unit  of  volume,  which 
is  the  density;  call  this  A,  and  the  above  becomes 

V=\J-.k (539) 

Denote  by  c  the  ratio  which  the  contraction  produced  in  a  given  vol 


MECHANICS     OF    MOLECULES. 


383 


ume  of  the  medium  by  the  pressure  of  a  standard  atmosphere  A,  bears 
to  the  volume  without  any  external  pressure;  then  will 


e  =  A  =  <7.i>„.30 
c  c 


tneKe*, 


.     .     .     .     (540) 


in  which  g  is  the  force  of  gravity  and  Dti  the  density  of  mercury  at 
a  standard  temperature. 

In  the  case  of  gases,  c  is  sensibly  equal  to  unity ;  for  if  such  bodies 
were  relieved  from  their  atmospheric  pressures  they  would  expand  in- 
definitely, thus  making  their  increments  of  volumes  sensibly  equal  \£ 
the  volumes  they  would  ultimately  attain. 


RELATION    OF    WAVE    VELOCITY    TO    WAVE    LENGTH. 

§  317. — Denote  the  resultant  displacement,  of  which  £,  r\,  and  £  are 
the  components,  by  tf;  and  the  angles  which  <r  makes  with  the  axes 
ar,  y,  and  2,  by  a,  j3,  and  y,  respectively;  then  will 

|  =  a  .  cos  a;     77  =  0".  cos  |3;     C  =  a  .  cos  y ; 
which,  substituted  in  the  second  members  of  Equations  (531),  give 


d~tJ 
dT2 

d? 


—  <f .  n,  .  cos  a, 

—  (f  .  wy2 .  cos  j3, 

—  tf  .  ?i,2 .  cos  y. 


J 


(541) 


Squaring,  adding,  taking  square  root,  and  denoting  the  resultant  by  em, 
we  have 

£2  =  (dll ?V+  /"^V+  (— ?V=  «rW   •  cos'a  +  n«   .  cos»/8  +  »\  .  cos'j-).    (542) 

The  first  member  is  the  square  of  the  resultant  acceleration  due  to  the 
molecular  action  developed  by  the  displacement  (f. 

Denote  by  a,,  (3t  ,  and  y,  the    angles    whict    the    direction    of  this 


384 


ELEMENTS    OF    ANALYTICAL    MECHANICS, 


resultant    makes    with    the    axes  a-,  y,  and   z,  respectively ;    and    by    \f 
the  inclination  of  this  direction  to  that  of  displacement.      Then  will 

cos  tp  =  cos  a  .  cos  ct/  -f-  cos  (3  .  cos  fif  -\-  cos  y  .  cos  yt  .     .  (543) 

The  components  of  the  acceleration,  in  the  directions  of   the  axes  ar,  y, 
and  z,  are,  respectively, 

sm  cos  a{ ;     €m  cos  /3t ;     em  cos  y/  ; 


and,  therefore,  Equations  (541), 


em  cos  a; 


Whence, 


em  cos  ft  =  - 

e„  cos  y,  =  — 

COS  OLt   =  — 

cos  (3t  =  — 

cos  y;  =  — 


<f  .  nj  .  cos  a, 
<f  .ny9 .  cos  /3, 
<S  .n? .  cos  y. 

<f  .nj .  cos  a 

(f  .  N  '  .  COS  j3 


3  .n? .  cos  y 


f 


(544) 


These,  in  Equation  (543),  give,  Equations  (531), 

em  .  cos  ijj  —  —  d  .  n*  =  —  d  .  (nj  .  cos2  a  +  ■/  •  cos2  0  -f  »,* .  cos2  y)  ; 
and  replacing  n^  nx,  ny,  and  rc2,  by  their  values,  Equations  (529), 

V*        V2  V*  V* 

-71  =  yr  *  cos  a  +  Tj  '  cos  P  +  Ti  *  cos  7- 

A  Aj,  A~  A^ 

But,  because  the  number  of  waves,  in  a  unit  of  time,  arising  from  the 
components  of  a  common  initial  disturbance  must  be  the  same,  tiie 
soefficients  of  the  circular  functions  above  must  be  equal,  and  hence, 


V*         y*  y* 

-A  =  -51  (cos2  a  +  cos2  0  +  cos2  y)  =  y^- 

A^  A#  Ax 


(545) 


Whence  the  wave  velocity  is  proportional  to  the  wave  length. 


MECHANICS    OF     MOLECULES. 


385 


SURFACE    OF    ELASTICITY. 


§  318. — Replacing,  in  Equations  (541),  nx1  nyi  nx,  by  their  values 
in  Equations  (529),  multiplying  the  first  by  c  .it  kj .  m,  the  second 
by   c  .  7r  Ay2 .  m,  and  the  third  by  c  .  7r  Ar2 .  m,  we  liave 


i  2  rf2£ 

c  .  7r  .  A.  .  in  ,  -r— 2  = 


—  tf  .  c  .  4  7T3 .  m  F,2 .  cos  a, 


d2n 

c  .n  .  AK2  .m  .  -—  =  —  a  .  c  .  4  7r3 .  m  F/  .  cos  j9,    J,  .    (546) 


d2£ 
c  .  n  .  A,2 .  m  .  -— 2  =  —  <r  .  c  .  4  7T3 .  m  V* .  cos  y. 


Now,  7r .  Ax2,  7r .  Ay9,  and  tt  .  Ar2  are  the  projections  of  the  waves  arising 
from  the  component  displacements  £,  77,  and  £,  on  the  planes  yz,  xz, 
and  zy,  respectively;  and  if  every  molecule  in  each  of  these  waves  had 
the  same  acceleration,  the  first  members  would  measure  the  elastic  forces 
exerted  over  these  projections  by  making  c  equal  to  unity.  These  are, 
however,  not  equal ;  but  if  c  denote  a  proper  fractional  coefficient,  and 
f.„  €y,  and  ez  the  actual  elastic  forces  in  the  three  waves,  we  may  write, 


ex  =  —  <f  .  c  .  Vx2 .  cos  a, 
ey  =  —  a  .  c  .  Vy  .  cos  j3,    h 
tt  =  —  (f  .  c  .  V* .  cos  y. 


(547) 


m  which  c  =  4  c  .  7r3 .  m.     Squaring,  adding,  taking  square  root  of  sum, 
and  denoting  the  resultant  by  eff, 


eo  =  V  ***  +  *;  +  **  =  C.c.v/F#4.cos2a+Fy4 .  cos2  j3  +  F/ .  cos2y ; 

from  which  it  is  apparent  that  if  the  displacement  be  made  in  the 
direction  of  either  axis,  the  elastic  force  will  be  wholly  in  the  direction 
of  that  axis — a  property  possessed  by  these  particular  axes  in  conse- 
quence of  the  fact  that  they  were  assumed  in  directions  to  satisfy  the 
conditions  of  symmetry  in  molecular  arrangement,  which  caused  Equa- 
tions   (524)    to    reduce  to   Equations    (f25).      The    directions   of  these 


386 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


special   axes   are  called   axes  of  elasticity.      The   resultant    elastic    force 
will  not,  in  general,  act  in  the  direction  of  the  displacement. 

Denote  the  angles  which  e  makes  with  the  axes  of  elasticity  by  c^, 
Qt,  and  y,,  and  the  angle  which  it  makes  with  the  displacement  by  ^>, 
then   will 

cos  if)  =  cos  a  .  cos  ay  -f-  cos  j3  .  cos  (34  -f-  cos  y  .  cos  yt , 

e  .  cos  a{  =  ex  =  —  <f  .  c  .  F.2 .  cos  a, 

eo .  cos  j3,  =s  €y  =  —  a .  f  .  Vf  .  cos  j3, 

e  .  cos  y/  =  et  =  —  <r .  g .  V* .  cos  X. 

Whence 


cos  a/  = 


cos  j3/  = 


cos  y,  =  — 


<f 

•  f . 

F8 

cos  a 

e 

a 

» 

a 

?• 

cos 

0 

e 

a 

» 

a 

?• 

r  z     • 

cos 

r. 

,  ^ 


(548) 


which  substituted  above,  give, 

eo  .  cos  i/>  =  -  tf .  g .  F 2  =  -  tf  .  $•  ( F,2 .  cos8  a  -f  ^' .  cos*  j3  +  F,2 .  cos*  y) ; 

in  which  F    is  the  velocity  perpendicular  to  the  displacement.     Making 


we  have 


V=V;      Vx  =  a;      V-b;      V,  =  c; 


V=  Va2.  cos8a4-^2.  cos2j3  +  c8.cos8y    .     .     .     (549) 


The  quantities  a,  b,  and  c  are  called  definite  axes  of  elasticity,  in  con- 
tradistinction to  axes  of  elasticity  which  merely  give  direction.  The 
surface  of  which  the  above  is  the  equation,  is  called  the  surface  of 
elasticity.  The  value  of  F  will  measure  the  velocity  of  any  point  on 
the  wave  surface  in  a  direction  normal  to  the  displacement,  and  being 
squared  and  multiplied  by  a .  g  will  give  the  elasticity  de\  eloped  in 
the  direction  of  the  displacement  itself. 


MECHANICS    OF    MOLECULES.  387 

The  definite  axes  of  elasticity  are  the  geometrical  axes  of  figure  of 
the  surface  of  elasticity ;  the  general  axes  of  elasticity  are  directions 
parallel  to  these,  and  drawn  from  any  point  in  the  medium  taken  at 
pleasure. 


WAVE   SURFACE. 

§  319. — This  is  the  locus  of  those  molecules  which  have,  simulta 
neously,  the  same  phase,  §  309 ;  and  whatever  this  phase  may  be, 
the  particular  surface  characterized  by  it  will  be  concentric  with  that 
which  marks,  at  any  epoch,  the  exterior  limits  of  the  disturbance,  or 
upon  which  the  molecules  are  beginning  to  participate  in  the  disturb- 
ance propagation. 

It  is  now  the  question  to  determine  the  equation  of  this  latter 
surface ;  for  this  purpose,  assume  the  origin  of  co-ordinates  at  the 
point  of  primitive  disturbance,  and  let 

Ix  +  my  +  nz  =  V  .     .    .     .     .    .     (550) 

be  the  equation  of  a  plane  tangent  to  the  wave  front  at  any  point, 
and  at  the  end  of  a  unit  of  time.  The  coefficients  /,  m,  and  n,  will 
be  the  cosines  of  the  angles  which  the  normal  to  this  plane  makes 
with  the  axes  xyz,  respectively,  and  its  length  will  measure  the 
velocity  F,  of  wave  propagation  in  its  own  direction.  This  plane  must 
be  parallel  to  the  displacement  and  its  normal  perpendicular  thereto; 
hence 

I  cos  a  -f  m  cos  ]3  +  n  cos  y  =  0     .     .    .     .    (551) ; 

also 

cos9  a  +  cos*  (3  -f  cos9  y  =  1     .     .     .     .     (552). 

Equations   (549),  (550),  (551),   and   (552)   must  exist  simultaneously 
for   real   values   of  the   cosines   of  a,  /3,  and   y.     To   find   an    equation 
which   shall  express  this  condition,  square  Eq.  (549),  and  divide  it  by 
V*  •  cos9  a,  it  becomes 


388  ELEMENTS    OF    ANALYTICAL    MECHANICS. 


,        COS2  0      ,  COS*  y 

COS"1  a  COS2  a 

(558) 


V*  COS2  a 

divide  Eq.  (551)  by  cos  a,  we  have 


eogij    ,         cos  y        m  /e*,x 

l  +  m +  n =  0 (554); 

COS  a  COS  a 


and  divide  Eq.  (552)  by  cos2  a,  the  result  is 


COS2/?  COS«y  1 

1  H H =  .      (555). 

COS2  a  COS2  a  COS2  a 


Equations  (553)  and  (555)  give 


,    ..    cos2  B            cos2  > 
a2  -|-  b* +  c' - 


cos2  B        cos2  y  cos2  rt  cos2  a 

"cos2  a  ~*~  cos2  a  ~~  F2  "  ' 

whence 

r*i-«a+(F2-&2).<^-^  +  (F2-c2).^-?  =  0        ....    (556). 

'  '    cos2  a  COS2  a  V 

From  Equation  (554)  we  have 

cos/3 

f  -f-  rn 

cos  y  cos  a 

cos  a  *t  ' 

which  in  Equation  (556)  gives 

r(r2-62)n2+(F2-c')m2].^— +2(F2-c2)./-OT.^-^  =  -(F2-a2)n2-(F2-c»)/2r 
1  Cos2  a  COS  a 

or 

•  I 

eos20  ( F2  —  c2) .  /  .  m  cosfl  _         ( F »  —  a2)  n2  -f(  F2  —  c2)  P 

^r;  +  2  ( V1  -  ¥)  n?  +  {V*  -  c*)m* '  cosl  " "       (  F2  —  62)  n2  +  (  F2  -  c2)ro2  * 

cos  j3 
and  solving  with  respect  to  ,  there  will  result, 

• °  r  cos  a 


cos  0  _       (r»-c»).m.lTw^-[(F»-a»)(  F2  -6a)»«+(r»  -aa)(  Fa  -c*)m*+(  F2  -  c2)(F2  -  &2)i»] 

iosl  ~  ~  <  F*-6»)  »a + (  F»  -c«)  «i» 

(557,; 


MECHANICS    OF    MOLECULES.  389 

and  this  in  Equation  (554)  gives 

cos  y  _      (J™-b*).nJ±my/^{V*-a*)(V*-b*)?i*+(V*-a*){F*-c*)m*+(V*--c*){r*--b*)l*] 
e^~~  (P»-&*)n»  +  (F»-c«)m* 

(558). 

For  any  assumed  displacement,  the  value  of  F,  Eq.  (549),  becomes 
known,  and  the  values  of  the  first  members  of  Eqs.  (55  7)  and  (558) 
must  be  real;  whence  /,  m,  and  n,  must,  in  addition  to  Eq.  (549), 
also  satisfv  the  condition 

(  V'  —  a2)(  F*  -  62)-2  -f  {  V*  -  «•)(  F2  -  c2)wi2  +  (  F2  -  62)(  K»  -  c2)/2  =  0. 

Dividing-  by 

(  F2  -  a2)  ( 72  -  62)  (  V%  -  c2), 

and  inverting  the  order  of  the  terms, 

P  m*  n1       _ 

|Tt ai  "T"  jrs  _  £2  +   jT2  __  c«  —  0-    •     •     •     •     (559) 

From  this  equation,  together  with  Equation  (550),  and  the  relation 

P  +  m2  +  n2  =  1, (560) 

we  have  all  the  conditions  necessary  to  find  the  equation   of  the  wave 
surface ;   this  is  done  by  eliminating  V,  m,  /,  and  n. 

For  this  purpose,  differentiate    each    of  these  equations  with  respect 
to  the  quantities  to  be  eliminated.     We  have,  from  Equation  (550), 

(1) xdl  +  ydm  +  zdn  =  d  V; 

from  Equation  (560), 

(2) ldl-\-mdm  +  ndn  =  0'1 

and  from  Equation  (559), 

Idl         '    mdm      ,      ndn 


(3) 


Idl       ,     mdm  ndn    _  _.  .  v  / P  m*  w»       \ 

V*~^a?  "*"  V*  -  #>  "*"  V*  -  c2  =:  \(V*-a*)*  ""*"  (P»-*)J  "*"  (  F«  _  <<*")i/' 


Multiply  the   first   by  A,  the   second    by    —  A',   tlie   third   by  —  1,  and 
add    members   to    members,  and  collect   the    coefficient?    of  l4ke  differ- 
entials ;   there  will  result, 

24 


390 


ELEMENTS     OF    ANALYTICAL    MECHANICS, 


(xx-X'l  -^l—.^dl 
+  \Xy-X'm-  v™_bi}d 


m 


n 


Hx-r\w^ 


=  0. 


/    TTO  19.9 


(V*-b2f   '   (V- 


Taking  X  and  X'  of  such  values  as  to  make  the  coefficients  of  d  V  auci 
in  each  zero,  the  equation  will  reduce  to  the  first  two  terras;  and  as 
dm  and  dl  are  wholly  arbitrary,  Equation  (560),.  as  long  as  dn  is 
undetermined,  we  may,  from  the  principle  of  indeterminate  coefficients, 
write, 


<*) 


(5) 


(6) 


I 


•     '-~       "•           7* -a2      '7 

1  u_,  Vm                           —  o 

•     A  y       A  m        yi  _  p  —  u» 

JLs     -  x'n  —                   —  0 

.       A  *           AW              J^2  _  c2   —    U» 

2      ri. 

Z2                      m8                      n* 

i                         I. 

Multiply  (4)  by  I,  (5)  by  m,   (6)   by  w,  add    and   reduce  by  Equation* 
(550),  (560),  and  (559);    we  have 


(8) 


aF-a'=0; 


Multiply  (4)  by  x,  (5)  by  y,  and  (6)  by  z;  add  and  reduce  by  Equa- 
tion (550)  and  the  relation  a*8  +  y2  +  z2  =  r* ;  we  have 


Xr         {XV+——  +  ——  + 


nz      \ 

yr-i .)  =  o; 


substituting  for  a'  its  value,  (8),  and  transposing, 


(9)      .     .     a  (r»-  F5)  = 


/#  my 


"*"     V*  A3  "T"     172 


V>  _  tf»    '     JTi  _  a 


,8  » 


MECHANICS     OF     MOLECULES.  391 

transposing  in  (4),  (5),  and  (6),  squaring  and  adding,   we  have 

^  H  -  1"  4-  —1 + 5l_  4- n——  • 

T  j v2  -  ay  \  (v*  -  by  T  (F2  -  c>f ' 

substituting  for  X'1  its  value,  (8),  and  reducing  by  (7),  we  have 


jy(r._  H  =  ^; 


and,  therefore, 


(10)     •      •      •      •      ^»  —  TTTIs         lzn  i       ^'  —  "3 


V  (r8  -  V2)  '  ""  H  -   F8' 

Substituting  these  in  (4),  we  find 


_x_ /       1  1       \ 

TZT\nj  ~     \rrZTv>  +  V"  -  a)  ' 


whence 


similarly, 


F(r8 

FZ 


r*  -  o}  ~  V3  -  a2 ' 

r*'l'^  =  F8  -  62 ; 
z  V  n 


r2-c*~  F8  -  c* ' 

multiply  the  first  by  #,  the  second  by  y,  the  third  by  2,  add  and  re- 
duce by  (9)  and  (10) ;  we  have 

x*  y*  z% 

~1 i"  +  "1 Ta  +  1 2  =  1 (56^) 

r8  — -  ar        r2  —  b2        r8  — -  r 

From  this,  which  is  one  form  of  the  equation  of  the  wave  surface,  sub- 
tract 


and  we  have 


x%  4-  y%  +  s'      , 


a8*8  &V  cV  /-B_„ 

3 i  +  -r-hi  +  j r  =  o   .    .    .    .    (562 

r1  —  a8        r8  —  6"         r*  —  r 


which  is  a  second  form  of  the  equation  of  the  wave  surface. 

Clearing  the  fractions,  it  becomes,  after  substituting  for  r*  its  value 
**  +  yf  +  •*, 


392 


ELEMENTS    OF    ANALYTICAL    MECHANICS 


(*«  +  f  +  z2)  («2  x*  +  i2  f  +  c2  z2)  ^| 

-  a8  (62  +  c2)  *" 

-  &9(a2  +  c*)y2 

-  c2  (a2  +  62)  sl 
+  a2  b2  c* 


,  =  0 .     .     .     (563) 


DOUBLE    WAVE    VELOCITY. 

§  320. — The  radius  vector  r  measures  the  velocity  of  the  point  ol 
the  wave  to  which  it  belongs;  and  denoting  by  lt,  m,,  and  nt  the 
sosines  of  the  angles  which  r  makes  with  x,  y,  and  0,  respectively, 
we  have 

x  ==  r  .  lt ;     y  r=  r  mi ;     2  se  f  *, ; 

and   writing  Vr  for  r,  we  have,  by  substituting  in  Equation  (563),  and 
dividing  by  VrA .  a2 .  62 .  c2, 

a  trinomial  equation,  of  which  the  second  powers  of  the  equal  roots  are 

i =i(?+ ?) + *  (?  ~  i)  W  ■ A" ±  VTZrir' x  ^F^  (565) 

and  in  which, 


A'  =1 


a' 


V    ? 


+  », 


a' 


^1"=/,. 


62       a2 


—  n. . 


1 

1 

c2 

62 

1 

1 

?~ 

a2 

1 

1 

? 

62 

.     .     (566) 


/  i    i    '  /  i    i 

V      c2       a2  V      c2       a1 


;     .     ,     .     (567) 


If  a  >  b  >  c,  the  values  of  -4'  and  .4"  will  be  real,  and  there  will,  in 
general,  be  two  real  values  for  -j=-2;  and  with  this  condition,  Equation 
(565)  will    give    two   pairs  of  real  and  equal  roots  with  contrary  signs. 


MECHANICS   OF  MOLECULES.  393 

The   positive   roots   give    two   velocities   in    any   one   direction,   and    the 
negative  in  a  direction  contrary  to  this. 

Through  the  origin,  conceive  two  lines  to  be  drawn,  making  with 
the  axis  a,  angles  whose  cosines  are  a/  and  at) ;  with  the  axis  6,  an- 
gles whose  cosines  are  j3;  and  j3/y ;  with  the  axis  c,  angles  whose  co- 
sines are  yt  and  yu  ;   and  such  that 


J    (568) 


and  denote  the  angle  which  r  makes  with  the  first  of  these    lines  by 
«  ,  and  that  which  it  makes  with  the  second  by  ult ;   then  will 

A'  =  lt  at  +  nt .  yi  =  cos  ut , 
A"  =  l,**-,  —  nt  yJ    =  cos  ut). 

Vl  —  A'1  =  sin  ut ;     Vl  —  A!n  =  sin  uu . 
These,  in  Equation  (565),  give  for  the  two  values  of  -=-4, 

|Ti  =  i  [f  +  J)  +  i  (->  ~  ^)  •  (cos  ui  • cos  u»  +  sin  ** ■ sin  M><)  •  •  (569) 

T"»  =  *  (?  *  a"8)  +  *  (?  ~  <?)  *  (C0S  *' '  C°S  ""  "  S'n  *'  *  Sln  W//)  '  '  (570) 

» 

and  by  subtraction, 

^"«  -  ITi  =  (?  "  3)  •  sin  tt' "  ™n  tf"  '     *     '     '  (5V1) 

Now, 

y     and   -^r 


'i  f* 


are  the  retardations  of  wave  velocity.  As  long  as  a  and  c  differ,  the 
second  member  can  only  reduce  to  zero,  when  ut  or  uu  is  zero ;  whence 
it   appears  that,  as  a  general   rule,  every  direction   except  two  ia  distin- 


394 


ELEMENTS    OF    ANALYTICAL    MECHANICS, 


guished  by  transmitting  two  waves,  one  in  advance  of  the  other.  The 
two  directions  which  form  the  exceptions  are  in  the  plane  of  the  axes 
of  greatest  and  least  elasticity,  and  make  with  these  axes  the  angles 
of  which  the  cosines  are  at  and  y.,  a.t{  and  y^,  Equations  (568).  In 
these  directions  the  waves  will  travel  with  equal  velocities. 

Any  direction  along  which  the  component  waves  travel  with  equal 
velocities  is  called  an  axis  of  equal  wave  velocity.  All  bodies  in 
which  the  elasticities  in  three  rectangular  directions  differ,  possess, 
Equation  (5*71),  two  of  these  axes,  and  are  called  biaxial  bodies.  Tin- 
retardation  of  one  component  wave  over  that  of  the  other,  will  vary 
with  the  inclination  of  the  direction  of  its  motion  to  the  axis  of  equal 
wave  velocity;  and  Equation  (571)  shows  that  the  loci  of  equal  retarda 
tions  will  be  arranged  in  the  form  of  spherical  le7nniscates  about  the 
poles  of  the  axes. 

§  321. — The  form  of  the  wave  surface    and    its    properties     become 
better  known  from  its  principal  sections  and  singular  points. 
Its  sections  by  the  planes  yzy  xz,  and  xy  give,  respectively, 


*  =  0  ;     (y*  +  zx  -  a2)  (62  y2  +  c2  z*  -  V  c2)  =  0, 


y  =  0;      (z*  +  x%  -  62)  (c2  z1  +  a2*2  -  c%  a>)  =  0,  "fj)    \    .    (572) 


*  =  0;     (a'  +  y'-c2)  (aV  +62y2  -  a2  62)  =  0, 


If  a  be  greater,  and  c  less  than  6,  then  will  the  first  give  a  circle 
and  an  ellipse,  the  latter  lying  wholly  within  the  former ;  the  third 
will  give  the  same  kind  of  curves,  but  the  ellipse  will  wholly  envelop 
the  circle ;  the  second  will  give  the  same  kind  of  curves,  intersecting 
one  another  in  four  points.  This  last  is  the  most  important.  It  is  the 
section  parallel  to  the  axes  of  greatest  and  least  elasticities, 

9 

§  322.— If  b  =  c,  then,  Equations  (568), 


a,  =  1 ;     7,  =  0  : 


MECHANICS    OF    MOLECULES.  395 

the  axes  will  coincide  with  one  another  and  with  the  axis  a,  that  is, 
with  x\    ut  will  equal  utl,  and,  Equation  (571), 

Also,  Equation  (563), 

(x2  +  y*  +  z2  -  c8)  [a5  g  +  c2  (y2  +  z1)  -  a2  c2]  =  0  .     .     (574) 

and  the  wave  surface  will  be  resolved  into  the  surface  of  a  sphere,  and 
that  of  an  ellipsoid  of  revolution.  Making  u/  =  0,  it  will  be  seen 
from  Equation  (571)  that  these  waves  travel  with  equal  velocities  in 
the  direction  of  the  axis  a.  For  anv  other  value  for  u.  since  u.  =  u  , 
cos  u4  cos  uu  +  sin  ut  sin  uu  =  1,  Equations  (569)  and  (570)  become 

^  =  7-;     f^.  =  ?■-(■?■- -yj  •  «« f«/.-    •    (^) 

and  it  hence  appears,  that  the  relocity  of  one  of  the  component  waves 
will  be  constant  throughout  its  entire  extent,  while  that  of  the  other 
will  be  variable  from  one  point  to  another.  The  first  is  called  the  or- 
dinary, the  second  the  extra-ordinary  wave. 

If  c  be  greater  than  a,  then  will  the  ellipsoid  be  prolate ;  if  less 
than  a,  it  will  be  oblate.  There  is  but  one  direction  which  will  make 
Vr  2  =  Vr  2,  and  that  is  coincident  with  the  axis  a.  Bodies  in  which 
this  is  true  have  but  one  axis  of  equal  wave  velocity,  and  are  called 
Uniaxial  bodies. 

From  Equation  (571)  it  appears  that  the  loci  of  equal    retardations 
are  concentric  surfaces,  of  which  the  common  axis  is  on   the  axis  of 
equal  wave  velocity,  and  common  vertex  at  the  origin. 

UMBILIC    POINTS.  9 

§  323. — Let  L  =  0  represent  Equation  (563),  and  take 

cos  A  = —  ;     cos  B  = r—  ;     cos  C  =  —  -7— ;       (5 !  6) 

w     dz  w     ay  w     dx 

in  which  J,  B,  and  C  are  the  angles  ivhich  a  tangent  plane  to  the  sur 


396 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


face    makes  with    the    co-ordinate    planes  xy,  xz1  and  y  z,  respectively 
and, 

l_  1 

w 


J idLV1  ,    /dL\*      (dL\l 

M*)+fo)+U7) 


.     .    .    .     (577) 


Performing  the  operation  here  indicated  on  Equation  (563),  we  have 


d_L 
dz 


Z-^=2z(aixt  +  b*y'  +  c2  z2)  +  2  c*  z  (x>  +  y*  +  z*  -  a*  -  b% 


dL 
dy 

d_L 
dx 


fL  =  2  y  (a8  x*  +  6*  y*  +  c2  z?)  +  2  6s  y  (x*  +  f  +  z8  -  a8  -  c!) ; 


=  2  z  (a8  a;8  +  62  y9  +  c2  z2)  +  2  a2  z  (z2  +  y8  +  z2  -  62  -  c') . 


Making   y  =  0,  brings  the   tangential  point    in    the    plane  a  c,  and   the 
above  become 


dL 

dz 

dL 

dy 

dL 
dx 


=  2  z  (a8  x>  +  c2  2s)  +  2  c2  2  (z8  +  z2  -  a2  -  fr2), 


0, 


=  2  *  (a8  ^  +  *' z2)  +2ola;(^  +  2,-Jf-4 


f.    .    .     (578) 


the   second  of  which    shows    the    tangent   plane    to   be    normal  to  the 
plane  a  c. 

But  y  =  0  gives,  Equations  (572),     - 

x*  +  z8  -  &*  =  o  ;     a2  z8  +  c8  z2  —  a8  c2  =  0, 


whence  we  have 


z  = 


ar  — i  | 


I 


-*Vf? 


-ft2 


a'  —  c 


(579) 


for  the  co-ordinates  of  the  points  in  which  the  circle  and  ellipse  inter 


MECHANICS    OF    MOLECULES 


397 


sect,  and  which  are  real    as   long  as  a  >  b  >  c.      Substituting  these  in 
Equations  (576),  (577),  and  (578),  we  have 

cos  A  =  - ;   cos  B  =  - ;    cos  C ■  =  - ; 

hence  the  points  of  intersection  of  the  ellipse  and  circle  in  the  plane 
of  the  axis  a  c,  are  the  vertices  of  couoidal  cusps,  each  having  a  tan- 
gent cone.  If  a  line  be  drawn  tangent  both  to  the  ellipse  and  the 
circle  in  the  plane  a  c,  the  tangential  points  will  belong  to  the  cir- 
cumference of  a  circle  along  which  a  plane  through  this  line  may  be 
drawn  tangent  to  the  wave  surface.  This  circumference  is  in  fact  the 
margin  of  the  conoidal  or  umbilic  cusp,  determined  by  the  surface  of 
the  tangent  cone  reaching  its  limit  by  becoming  a  plane  in  the  grad- 
ual increase  of  the  inclination  of  its  elements,  as  the  tangential  cir- 
cumference recedes  from  the  cusp  point.  A  narrow  annular  plane 
wave,  starting  from  this  circle,  will  contract  to  a  point  in  one  direc- 
tion ;  and,  conversely,  an  element  of  a  plane  wave  starting  in  the  op- 
posite direction  will  expand  into  a  ring. 

It  thus  appears  that  the  general  wave  surface,  and  of  which  (563) 
is  the  equation,  consists  of  two  nappes,  the  one  wholly  within  the 
other,  except  at  four  points,  where  they 
unite,  and  at  each  of  which  they  form 
a  double  umbilic,  somewhat  after  the 
manner  of  the  opposite  nappes  of  a  very 
obtuse  cone.  The  figure  represents  a 
model  of  the  wave  surface,  so  cut,  by 
three  rectangular  planes,  as  to  snow  two 
of  the  umbilic  points,  as  well  as  the 
general  course  of  the  nappes,  by  the  re- 
moval of  a  pair  of  the  resulting  diedral 
quadrantal  fragments. 


MOLECULAR    VELOCITY. 


§  324,— Multiply  the  first  of  Equations  (531)  by  2  d  |,  the  second 
by  2drj,  the  third  by  2  d  £  and  integrate ;  there  will  result,  recollect- 
ing that  the  molecule  is  moved  from  its  place  of  rest 


398  ELEMENTS    OF    ANALYTICAL    MECHANICS. 


dp 

df  "~ 

-  <  .  ?, 

drf 
df  ~ 

-  K  •  n\ 

d? 

df  ' 

-n\.?. 

(580) 


whence  it  appears  that  the  velocity  of  a  molecule  in  the  direction  ol 
either  axis  is  proportional  to  its  displacement  in  that  direction,  from 
its  place  of  rest.  The  place  of  rest  is  only  relative.  When  a  mole- 
cule is  in  a  position  such  that  its  neighbors  are  symmetrically  disposed 
around  it,  it  is  in  its  place  of  rest,  and  its  displacement  therefrom  will 
be  directly  proportional  to  the  excess  of  condensation  on  one  side  over 
that  on  the  other.  This  excess  and  the  molecule's  motion  will  reduce 
to  zero  simultaneously,  and  a  single  displacement,  not  repeated,  can 
only  give  rise  to  what  is  called  a  pulse. 

These    equations   also    show    that   the    living  force  of  the  molecule  is 
proportional  to  the  square  of  the  displacement. 

MOLECULAR    ORBITS. 

§  325. — The  molecular  orbits  are  on  the  wave  front.  Suppose  the 
wave  due  to  the  displacement  f  to  be  superposed  upon  that  due  to  77, 
and  take  a  molecule  of  which  the  place  of  rest  is  on  the  axis  r. 
The  first  and  second  of  Equations  (528),  will  be  sufficient  to  find  the 
orbit  of  this  molecule  under  the  simultaneous  action  of  both  waves. 
From  these  two  equations  we  find,  ajfer  writing  z  for  rm  and  rr, 

(1)  .     .     .     .    ^.(Vx.t-z)  =  sirr1^-, 
At  a, 

(2).     .     .     .     ^.(P^-^sin-1-! 


(3).     .    .     .     ^  .  {Vm  .  t  -  z)  =  cos-'l/l  -  £, 


a. 


(4).     .    .     .     y^.(Vy.t-z)  =  cor1Vl-£. 


af 


MECHANICS    OF    MOLECULES.  399 

Subtracting  (2)  from  (1), 


rr  /  V>  A  -  Z  Vy  1  -   Z\ 


V,.t  —  z       V„ .  t  —  z\         .  _i  £         .  _,  rt 

2  7r  I  -    -7-  — —     -1  =  sin sin      — ; 

a,  a 


in  which  Vm .  t  «—  t,  is  the  distance  of  the  wave  front  due  to  g  from 
the  molecule's  place  of  rest,  and  Vy .  t  —  z,  that  of  the  wave  front  due 
to  r\  from  the  same  point.     Make 

t,  =  time  required  for  the  wave  front  due  to  £-to  travel  over  Vm .  t  —  z\ 

u  »f  •>    . 

A,, 

?/  "        Vy.t-z; 

Ay, 


r.= 

u 

M 

M 

^  = 

u 

u 

M 

ry  ~ 

u 

u 

U 

then  will 

F. 

.t 

—  0 

1 

2. 

i 

which  substituted  above  gives,  after  taking  cosine  of  both  members, 


Clearing  the  radical  and  reducing, 

—4  +  -^  -  2  cos  2  7T  — £  .  -L .  Jl  —  sir.2  2  ?r  — ^  =  0  .     .     (582) 
«.         V  1-,     «.     «y  T, 

which  is  the  equation  of  an  ellipse  referred  to  its  centre. 

§  326. — To  find  the  position  of  the  transverse  axis,  take  the  usual 
formulas  for  the  transformation  of  co-ordinates  from  one  set,  which  are 
rectangular,  to  another,  also  rectangular.     They  are, 

|  =  %'  cos  (p  —  7]'  sin  <p, 
7]  =  %  sin  <p  +  7\'  cos  <p ; 

in  which  <p  is  the  angle  which  the  axis  %'  makes  with  that  of  £ 

Substituting*  these    values  of  £  and   t\  in   Aquatic  n  (582),  collecting 


400  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

the   coefficients,  and  placing  that  of  the    rectangle  £'  r\ ',  equal  to  zeuo 
we  have 

2  sin  9  .  cos  <p  (a,9  —  of)  —  2  (sin2  <p  —  cos2  9)  .  a,  .  ap .  cos  2  tt  —  =  0 ; 


T, 


and  because 

sin2  <p  —  cos2  9  =  cos  2  <p, 

2  sin  9  .  cos  <p  =  sin  2  <p. 


the  above  becomes, 


tan  2  <p  =  2  .  -~^  .  cos  2  tt  .  —     .     .     .     .     (583) 

'Ve     ~~"    ™«  '   # 


*  y  '  / 


§  32*7. — Now,  if  the  successive  pairs  of  component  waves  which  dis- 
turb  the   molecule,  reach  it  with  a  variable    difference    of  phase,  then 

will  cos  2  n  —  be  variable,  and  the  transverse  axis  of   the  elliptical  or- 

bit  be  continually  shifting  its  place.  A  wave  in  which  the  molecular 
motions  fulfil  this  condition  is  called  a  common  wave;  being  far  the 
most  frequent  in  nature.  When  the  successive  pairs  of  component 
waves  are  such  as  to  make  the  second  member  of  Equation  (583)  con- 
stant, the  transverse  axes  of  the  molecular  orbits  will  retain  the  same 
direction,  and  the  wave  is  said  to  be  elliptically  polarized. 

§  328. — If  —  equal  J,  or  any  odd  multiple  of  ^,  and  ax  =  a^  then 
will,  Equation  (582), 

e  +  rf  -«,2  =  0, (584) 

and  the  orbit  becomes  a  circle.  When  this  happens,  the  wave  is  said 
to  be  circularly  polarized. 

§  329. — If  —  be  equal  to  any  even  multiple  of  1,  then  will 


T/ 


cos  2  n .  -i-  =  1 ;      sin2  2  7r  •  —  =  0  ; 

T  T 


*nd,  Equation  (582), 


=  0, ...     (685) 

a,        ar 


MECHANICS    OF    MOLECULES.  401 

and  the  orbit  is  a  straight  line  through  the  molecule's  place  of  rest 
The  motion  of  the  molecule  wil  take  place  in  a  plane  normal  to  the 
wave  front,  and  the  wave  is  said  to  be  plane  polarized ;  and  a  plane 
normal  to  the  wave  front  and  in  the  molecular  paths,  is  called  the 
plane  of  polarization. 

§  330. — Referring    the    curve    to    the    new    axes,   and    omitting   the 
accents  from  %'  and  rj',  Equation   (58*2)   may  be  written, 

— 8  +  — 2  -  sin2  2  n .  -*-  =  0, (586) 

ax         av  r4 

ill  which  a,  and  ay  will  take  newr  values. 


REFLEXION    AND    REFRACTION    OF    WAVES. 

§  331. — The  elastic  force  which  the  molecules  in  the  surface  of  one 
body  exert  upon  those  in  the  surface  of  another,  in  sensible  contact, 
must,  when  the  molecules  are  at  relative  rest,  be  equal  to  that  exerted 
by  the  molecules  in  the  interior  of  either  body ;  else  these  surface 
molecules  would  be  urged  in  opposite  directions  by  unequal  forces,  and 
relative  repose  would  be  impossible.  But,  for  equal  displacements,  the 
elastic  forces  developed  in  different  bodies  are  in  general  unequal,  and 
this  is  one  of  the  most  common  of  the  causes  that  produce  a  resolu- 
tion of  primitive  into  secondary  or  component  waves. 

The  velocity  of  a  wave  molecule  varies,  liquations  (580),  directly  a» 
the  molecule's  distance  from  its  place  of  rest.  If,  therefore,  a  wave,  in 
its  progress  through  any  medium,  meet  with  a  constitutional  change  of 
elasticity  or  density,  the  elastic  force  developed  at  the  place  of  change 
will  either  be  greater  or  less  than  that  which  determined  the  places  of 
rest  in  the  interior  of  either  body.  In  the  first  case,  the  condensation 
in  front  cannot,  by  the  forward  movement,  reduce  to  an  equality  with 
that  behind ;  the  surface  molecules  will  first  be  checked,  and  then  partly 
driven  back  upon  those  behind,  and  a  return  and  an  onward  pulse  will 
proceed  in  opposite  directions  from  the  surface  which  marks  the  change 
of  structure,  as  from  a  primitive  disturbance.  In  the  second  case,  the 
molecules,   meeting   with   less  opposition,  will    go  beyond    their   neutral 


402  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

limits  with  reference  to  those  behind,  the  latter  will  close  up  in  sue 
cession,  and  thus  a  return  and  transmitted  pulse  will  arise  as  before, 
but  with  this  difference,  viz. :  in  the  latter  case,  the  molecular  motions 
in  the  return  pulse  will  continue  in  the  same  direction  as  before, 
whereas,  in  the  former  case,  those  motions  will  be  reversed.  The  retui  n 
pulse  is  said  to  be  reflected ;  that  transmitted,  refracted.  The  primitive 
pulse,  and  of  which  these  are  the  components,  is  called  the  incident 
pulse.  A  change  of  density  or  of  elasticity  will,  Equation  (537),  pro- 
duce a  change  in  the  velocity  of  wave  propagation.  A  surface  which 
is  the  locus  of  a  change  of  density  or  of  elasticity,  is  called  a  deviating 
surface.  Two  planes  which  are  tangent,  the  one  to  the  deviating  sur- 
face, the  other  to  the  wave  front,  at  a  point  common  to  both,  will 
intersect  in  a  line  parallel  to  that  of  the  nodes  of  the  molecular  orbits, 
which  are  in  the  deviating  surface  and  near  the  common  tangential 
point.  This  line  of  intersection  is  called  the  line  of  nodes.  A  plane 
through  the  tangential  point  and  perpendicular  to  the  line  of  nodes,  is 
called  the  plane  of  incidence.  The  medium  through  which  the  wave 
moves  before  it  meets  the  deviating  surface,  is  called  the  medium  of 
incidence;  that  into  which  it  enters  on  passing  this  surface,  the  medium 
of  intr  omittance. 

§  332. — Let  i  be  a  point 
common  both  to  the  wave  and 
ieviatinor  surface.  A  C  a  lin- 
ear  element  of  the  former,  and 
A  B  a  like  element  of  the  lat- 
ter, both  lying  in  the  plane 
of  incidence.  Denote  by  V 
and  X  the  velocity  and  length 

of  the  wave  in  the  medium  of  incidence ;  by  Vt  and  Xt  the  same  in  that 
of  intromittance ;  and  by  t  the  time.  Now,  supposing  the  wave  to  proceed 
in  the  direction  C  B,  and  taking  A  B  =  d  s,  we  have  CB  =  V.dt. 

But  while  the  point  C,  in  the  incident  wave  front,  is  moving  from 
C  to  B,  the  reflected  pulse,  proceeding  from  A  as  a  centre  of  disturb- 
ance, will  move  over  a  distance  equal  to  V  d  t  in  the  medium  of  inci- 
dence; the  refracted  pulse  over  a  distance  equal  to  V{.dt   in    that    of 


MECHANICS    OF    MOLECULES.  403 

intromittance.     With  A  as  a  centre,  and  radius  V.  d  /,  describe  the  arc 

««,  and  with  the  radius  V td  t,  the  arc  aV;  and  from  B  draw  the  tano-enU 

B  B  and  B  D ';  the  first  will  be  the  front  of  the  new  wave  element  in  the 

medium  of  incidence,  the  second  in  that  of  intromittance. 

§  333.— Denote  the  angle  C  A  B  =  A  B  B  by  9  ;  the  angle  A  B  B'  by 

9';  then  will 

d  s  .  sin  9  =  Vd  t ;     d  s  .  sin  9'  =  Vt  d  t  ,     .     .     .     (587) 

and  by  division,  denoting  the  ratio  of  the  velocities  by  m, 

sin  9         V  , 

,  =-77  =  ™ (088) 

whence  sin  9  =  m  sin  9' (589) 

The  angle  9  measures  the  inclination  of  the  incident,  and  ©'  that  of  the 
refracted  wave  to  the  deviating  surface.  These  are  equal,  respectively,  to 
the  angles  which  the  normals  to  the  incident  and  refracted  waves  make 
with  the  normal  to  the  deviating  surface,  at  the  point  of  incidence.  The 
first  is  called  the  angle  of  incidence,  the  second  the  angle  of  refraction. 
The  inclination  of  the  reflected  wave  to  the  deviating  surface,  is  called  the 
angle  of  reflexion.  The  normals  to  the  incident  and  reflected  waves  fall  on 
opposite  sides  of  the  normal  to  the  deviating  surface ;  and  because  the  ve- 
locity of  the  reflected  wave  is  equal  to  that  of  the  incident,  with  contrary 
sign,  Equation  (589)  becomes  applicable  to  the  reflected  wave,  by  making 
m  =  —  1. 

LIVING    FORCE    AND    QUANTITY    OF    MOTION    IN    A    PLANE    POLARIZED    WAVE. 

§  334. — Take  either  of  Equations  (528),  say  the  first,  and  which  relates 
,    to  a  wave  plane  polarized,  the  plane  of  polarization  being  perpendicular  to 
the  co-ordinate  plane  y  2,  differentiate  with  respect  to  g  and  t,  dropping 
•  the  subscripts — we  get 

-±=a.V.cos  —  (Vt-r)  — . 
d  t  X    v  '    A 

Denote  the  density  of  the  medium  by  A,  and  the  area  of  any  portion 
of  the  wave-front  by  a,  then  will  the  mass  between  two  consecutive  posi- 
tions of  this  area  be  a.A.rfr,  and  the  living  force  within  a  quarter  of  a 
wave-length  be 

t,.a.dr.-p-z=l         A.a.-^.a'.F*.cos*— (Vt-r)—  dr  =  i*K-.  A-«-  V.m* 
'+U  dvl   J  rt-r-n      X  AAA 


404  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

V 

Dividing  by  the  volume  a.V,  and  recalling  that  *  and  —  are  constant, 

A. 

we  shall  find  that  the  quantity  of  living  force  in  a  unit  of  volume  of  the 
medium  will  vary  directly  as  the  product  of  the  density  and  square  of  the 
greatest  displacement ;  and  the  relation  of  these  products,  in  the  case  of 
any  two  waves,  will  determine  the  relation  of  the  effects  of  these  waves 
upon  the  organs  of  sense  upon  which  they  act. 

Again,  the  quantity  of  motion  in  this  quarter  of  wave-length  will  be 


/•*  d?        s*Vt-r  =  o  2rr  2  ir 

A  .  a.  dr .—- = I  a  ■  a  .  a  .V.  cos  —  (Vt-  r)-—  dr=&.a.a.V 

r  +  ly  dt    J   Vt-r=i\  *  * 


(5»1) 


RESOLUTION    OF    LIVING    FORCE    AND    OF    MOTION,    BY    DEVIATING    SURFACES. 

§  335. — Take  the  co-ordinate  plane  xz  in  the  plane  of  incidence,  and 
the  axis  z  in  the  direction  of  the  normal  to  the  incident  wave,  the  axis  // 
will  be  parallel  to  the  line  of  the  nodes  of  the  molecular  orbit  in  the  devi- 
ating surface,  at  the  place  of  incidence.  Then,  preserving  the  notation  of 
§  332,  will  the  element  of  the  deviating  surface  at  the  place  of  incidence 
be  d  s  ,dyy  and  its  projections  upon  the  incident,  reflected  and  refracted 
wave-fronts,  respectively,  be  ds .dy . cos <p,  ds  .dy  cos p,  and  ds .dy  .  cos <p '. 
These  will  take  the  place  of  a  in  Equations  (590)  and  (591),  in  computing 
the  living  force  and  quantity  of  motion  in  the  incident,  reflected  and  re- 
fracted waves.  The  living  force  in  the  incident  must  be  equal  to  the  sum 
of  the  living  forces  of  its  reflected  and  refracted  components.  First  take 
the  wave  in  which  the  molecular  motions  are  parallel  to  the  axis  x,  and 
employ  the  subscripts  *,  r  and  t  to  denote  the  incident,  reflected  and  re- 
fracted or  transmitted  waves,  respectively.  The  living  force  in  a  quarter 
of  each  of  these  waves  will,  omitting  the  common  factors,  Equations  ^529), 
(545)  and  (590),  give 

A  .  cos  <p  .  V.  a2ar  4-  Ay .  cos  <p' .  Vt .  a?mt  —  A  .  cos  9  .  V.  a2,,-  =  0  ; 
or,  Equations  (588)  and  (589), 

,     ,  A     cos<p'   sin<p'     9         ,         "  ,„-„v 

«5.r+-r -•-— -•«'.«— af.«=0    .     .     .     .     (592) 

A    cos  <p    sin  <p 

in  which  A  and  A7  are  the  densities  of  the  medium  of  incidence  and  of 

intromittance. 

The  molecular  motions  are  all  parallel  to  the  plane  of  incidence,  and  at 

the  same  time  normal  to  the  directions  of  their  respective  wave  motions 


MECHANICS    OF    MOLECULES.  405 

they,  therefore,  make  with  one  another  angles  equal  to  those  made  by  the 
directions  of  these  latter  motions,  and  we  obtain  two  more  equations  from 
the  relations  of  Equations  (59)  for  the  resolution  and  composition  of  ob- 
lique forces.  The  angles  made  by  the  direction  of  the  motion  in  the  inci- 
dent with  the  directions  of  the  motions  in  the  reflected  and  refracted 
waves,  are  180°  —  2$  and  360°  —  (9  —  9'),  respectively  ;  and  the  angles 
under  which  the  directions  of  the  motions  in  the  latter  waves  are  inclined 
to  one  another,  is  180°  —  (9+9').     Whence 

„  T_  sin  (<p  —  <p') 

A.  COS  ©.  V.Uxr=   —  A  .  COS©  .  V.axi  . ; -  l 

sin  (9  -f  9  ) 

I     rr  tt  8m  2  9 

A  .  cos  9  .  V , .  axt  =  A  .  cos  9  .  V .  axi .  - — - — - — -  ; 

sin  (9  -f-  9  ) 

^«-^.*g£s3 (593) 

sin  (9  +  9  ) 

A     cos  9     sin  9         sin  2  9 
A;    cos  9     sin  9     sin  (94-9). 
Substituting  these  in  Equation  (592),  we  readily  find, 

A        4  cos2  9'.  sin8  9'       cos'q/.  sin2  9' 

Af  sin2  2  9  cos2  9  .  sin2  9  ' 

whence, 

/—         /—          sin  2  9         _    /-     cos  9.  sin  9  /*A«» 

Va/  =  Va-- .-i 7-Va. v   .      ,    .     .     (595) 

2  cos  9  .  sin  9  cos  9  .  sin  9 

Substituting  the  above  ratio  of  the  densities  in  the  equation  just  preced- 
ing, we  get 

2  cos  9'.  sin?'  , 

axt—axi- — : — -,—  -, — rr~  1 {ovb) 

sin  (9  -f  9  ) 

multiplying  this  by  Equation  (595),  member  by  member,  and  the  equa- 
tion giving  the  value  of  axr  by  V  A,  and  taking 

VA.a,-=l;      V^S«„  =  r;     v/A( . .  ax<  =  w, 
we  find 

sin  (9  —  9') 


v  = 


u  = 


sin  (9  -f-  9') 
sin  2  <p 


sin  (9  -f-  9') 
To  which  may  be  added  the  relations,  Equation  (589), 


(597) 
(506) 


,      sin  9  ,       ,/         sin2  9 

sin  9  = ;     cos  9=1/1 =- 

in  m2 

25 


406  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Transposing  the  term  of  which  ax<  is  a  factor  to  the  second  member  in 
Equation  (592),  subtracting  Equation  (593)  from  axl  =  axi,  dividing  the 
first  result  by  the  second,  and  multiplying  the  quotient  by  liquation  (593)', 
we  readily  find 

™        "  =  — —, (599) 

cos  .p  cos  © 

That  is,  the  projection  in  the  direction  of  wave  propagation  and  on  the 
deviating  surface,  of  the  greatest  displacement  in  the  incident,  increased 
by  that  in  the  reflected  wave,  is  equal  to  like  projeetion  of  the  greatest 
displacement  in  the  refracted  wave. 

Next,  take  the  wave  in  which  the  molecular  motions  are  parallel  to  the 
axis  y ;  these  are  parallel  to  the  deviating  surface.  The  motions  in  the 
incident,  reflected  and  refracted  waves  are  parallel  to  one  another,  and,  by 
the  principles  of  parallel  forces,  the  sum  of  the  motions  in  the  reflected 
and  refracted  waves  must  be  equal  to  that  in  the  incident.  The  equation 
for  the  living  force  will  be  the  same  as  before.  Whence  Equations  (529), 
(545)  and  (590),  omitting  the  common  factors, 

A  .  cos  ©  .  V.  a2yr  +  Ay .  cos  ©'.  Vt .  a2yt  —  A  .  cos  ©  .  V.  a'2yi  =  0  ; 

A  .  cos  ©  .V.  (tffr  -f-  A;.  cos©'.  Vt .  oiyt  —  A  .  cos  <p.V.  ayi  =  0    .     (600) 

In  which  A  and  A(  are,  as  before,  the  densities  of  the  medium  of  incidence 
and  of  intromittance,  respectively ;  or,  Equations  (588)  and  (589), 

0        A    sin©'    cos©'      .. 

a     -+- : -a    t—  a    •_  0 

A     sin  ©     cos  © 

A'   sin©'   cos<p' 
V+  —  •^-r--  --avt-ayi=0      .      .     .      (601) 

'       A    sin  ©     cos  ©      •  v       ' 

Transposing  the  terms  containing  ayr  and  ayi  to  the  second  members,  and 
dividing  the  first  by  the  second,  we  find 

<V  +  ^i  —  o~yt (602) 

That  is,  the  greatest  displacement  in  the  refracted  is  equal  to  the  sum  of 

the  greatest  displacements  in  the  incident  and  reflected  waves. 

A' 
Substituting  the  value  of  — ,  as  given  by  Equation  '597),  in  Equation 

(601),  we  have 


.    sin  ©  .  cos  ©  ■  ,nn  : 

sin  ©  .  cos  ©'      y         ' 


MECHANICS    OF    MOLECl'LES.  407 

Substituting  in  this,  first  the  value  of  «/,,,  and  then  of  a,/t.  deduced  from 
Equation  (602),  we  readily  get 

tan  (?  —  <p')  t 


tty  r  a»«  •  , 


«„,=  a. 


r  *  tan  (p  -f  <p') 
4  cos  <p'.  sin  <p' 


sin  2  <p  +  sin  2  9 


(604) 


(605) 


Multiplying  the  first  of  these  by  ^^  and  the  second  by  Equation  (595), 

and  making 

v'a  .  oryi  =  1  ;      >/*  ,  aff  =  *';     v^  .  ^  -=  u'; 
there  will  result, 


tan  (<p  —  <p') 
tan  (p  +  <p') 

sin  2  <p 


u 


(600) 


(607) 


sin  (p  +  9')  .  cos  (9  —  <p') 
§  336. — Divide  Equation  (598)  by  Equation  (597),  and  Equation  (607) 
by  Equation  (606),  replace  /•,  w,  v  and  *'  by  their  values,  and  substitute  for 
the  ratio  of  the  square  roots  of  the  densities  its  value  as  given  in  Equation 
(595),  we  find 


a*f 

cos  $ 

«»r 

.  cos  9 

*,t 

<v 

cos  p         sin  2  © 


'»» 


cos  <p     sin  (9  —  (p') 
sin  2  <p' 


(608) 


sin  (9  —  9')  .  cos  (9  +  9') 
But  a,, .  cos(p'  and  axr .  cos  9,  are  the  components  parallel  to  the  deviating 
surface  of  the  displacements  which  are  in  the  plane  of  incidence;  ayt  and 
ayT  are  already  parallel  to  the  deviating  surface;  whence,  as  long  as  <p><pV 
that  is,  as  long  as  the  velocity  of  wave-motion  in  the  medium  of  incidence 
exceeds  that  in  the  medium  of  iritromittance,  the  molecular  phases  in  the 
refracted  and  reflected  waves  will  be  opposite,  and  conversely. 

§  337. —  Denote  the  living  force  in  the  original  incident  wave,  sup- 
posed common,  by  unity ;  that  in  each  of  its  two  original  components 
will  be  denoted  by  one  half  of  unity,  and  the  total  living  force  of  the 
reflected  wave  will,  Equations  (597),  (606),  be 

■?."         *    sin*  (<p  4- <p')  ^*   tan2  (<p+<p')  K       ' 

and  that  of  the  refracted, 

B.  +  .^i(l-gi2^)+i(i-S^!)     (610) 

5  \         sin2  (?  +  ?)/       2  \         tair  (9  +  9  )/ 


408  ELEMENTS    OF    ANALYT    CAL    MECHANICS. 


POLARIZATION    BY    REFLEXION    AND    REFRACTION. 

§  338. — The  first,  term  in  the  second  member  of  Equation  (60„), 
measures  the  living  force  in  that  portion  of  the  reflected  wave  which 
1?  due  to  vibrations  parallel  to  the  plane  of  incidence ;  the  second,  that 
clue  to  vibrations  perpendicular  to  this  plane.  The  former  exceeds  the 
latter.  These  living  forces  being  proportional  to  the  squares  of  the 
greatest  displacements,  the  former  may  be  represented    by  a/,  and  the 

latter   by  ay2,  in  Equation  (582).      The    factor  — ,  in  this  equation,  de- 

termines  the  difference  of  phase  simultaneously  impressed  by  both 
waves  upon  the  same  molecule,  and  when  the  waves  have  passed  from 
one  medium  to  another,  its  value  will  depend  not  only  upon  the  na- 
ture of  both  media,  but  also  upon  the  action  'to  which  the  waves  may 
have  been  subjected  while  crossing  the  space  wherein  the  physical 
changes  occur  that  constitute  the  transition  from  one  medium  to  an- 
other. The  amount  of  this  action,  in  any  particular  case,  can  only  be 
known  from  experience.  The  resultant  waves,  both  in  the  medium  of 
incidence  and  of  intromittance,  will  be  elliptically  polarized. 

When  <p  -f-  <p'  =  90°,  then,  Equation  (589),  will  sin  cp'  =  cos  <p,  and 

sin  <p  .■  '     ' 

ra  = =  tan  cp  : i61 1) 

cos  9 

the  second  terra  of  Equation  (609)  will  disappear,  and  the  reflected 
wave  will  be  wholly  polarized  in  the  plane  of  incidence.  This  angle, 
of  which  the  tangent  is  equal  to  the  index  of  refraction,  is  called  the 
volarizing  angle. 

The  index  of  refraction  varies  with  the  wave  length,  Eqs.  (588),  (545), 
and  it  will,  therefore,  be  impossible  wholly  to  polarize,  by  a  single  re- 
flexion, a  wave  compounded  of  several  components,  having  different 
wave  lengths. 

Of  the  terms  of  tte  second  member  of  Equation  (610),  the  last  ih 
the  greater,  because 


sin5  (<p  —   p')       tan2  (<p  —  <p')    cos8  (cp  —  cp')  # 
sin2  (<p  -f-  cp')  '  '  tan2  (cp  +  cp')    cos2  (<p  +  y) 


'\  ' 


MECHANICS    OF    MOLECULES.  409 

and  the  excess  will  measure  the  preponderance  of  that  part  of  the  re- 
tracted wave  due  to  vibrations  perpendicular  over  that  due  to  vibra- 
tions parallel  to  the  plane  of  incidence.  This  excess  is  exactly  equal 
to  the  excess  in  the  reflected  wave  which  arises  from  vibrations  par- 
allel over  those  perpendicular  to  the  plane  of  incidence. 

§  338'.— If  the  wave  velocity  in  the  medium  of  incidence  be  less 
than  in  that  of  intromjttance,  then  will  m  be  less  than  unity,  and  the 
values  of  v  and  v'  become  imaginary  for  all  angles  of  incidence  greater 
than  that  whose  sine  is  equal  to  m,  and  at  this  limit  the  problem 
changes  its  nature.  In  fact,  this  is  the  limit  of  refraction,  according 
to  the  law  of  the  sines,  Equation  (589),  and  for  any  increase  of  the 
angle  of  incidence  beyond  this,  the  wave  will  be  wholly   reflected. 

§  339. — If  the  wave  be  plane  polarized,  and  its  plane  of  polarization 
inclined  to  that  of  incidence,  under  any  angle  denoted  by  a,  then  will 
the  reflected  component  displacements  parallel  and  perpendicular  to  the 
plane  of  incidence  be,  respectively,  Equations  (597)  and  (606), 

sin  (<p  —  <p')  ,      tan  (<p  —  <p')     . 

: — y- ■—  .  cos  a,  and ) — j-  .  sin  a. 

sin  (<p  -I-  <p  )  tan  (<p  ■+-  <p  ) 

The  component  waves  due  to  these  displacements  will  proceed  onwards. 

and    may   satisfy   the    condition   of  —    being   an    even    multiple    of   \ ; 

in  which  case  the  resultant  will,  Equation  (585),  be  a  plane  polarized 
wave.  Denote  the  inclination  of  its  plane  of  polarization  to  that  of 
reflexion  by  a',  then  will 

tan  (<p  —  <p')     . 

i       — r-z r  •  sin  a  i     ,     /\ 

v        tan  (<p  +  <p  )  cos  (<p  -f-  <p')  . 

tan  a'  =  -  =  - — y. ^ =  ) £  .  tan  a  .    (612) 

v        sin  (a  --  q/)  cos  (<p  —  <p  ) 

- — f ~  .  cos  a 

sin  (<p  -f  <p') 

If  9  +  <p'  =  90°,  then  will  a'  =  0°,  whatever  be  a ;  also  if  a  =  0°, 
then  will  of  =  0°  ;  finally,  if  <p  =  0°,  then  will  <p'  =  0,  and  a?  —  a. 
That  is,  when  a  plane  polarized  wave  is  incident  under  the  polarizing 
angle,  it  is  reflected  polarized  in  the  pkne  of  reflexion.  Where  an  in- 
cident wave  is  polarized  in  *l»<*  olanc  of    incidence,  the   reflected    wave 


410  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

preserves  its  plane  of  polarization  unchanged  under  all  angles  of  inci- 
dence. Finally,  under  a  perpendicular  incidence,  the  plane  of  polariza- 
tion of  the  incident  and  that  of  the  reflected  wave  coincide. 

Equation  (612)  shows  that  a'  is  always*  less  than  «,  and  that  the 
\  lane  of  polarization  approaches  that  of  incidence  at  each  reflexion,  and 
may  be  made,  by  a  sufficient  number  of  reflexions,  ultimately  to  coin- 
cide with  it. 

§  340. — Still,  supposing  the  velocity  of  the  wave  less  in  the  medium 
of  incidence  than  in  that  of  intromittance,  or  9'  >  <p  ,  let  the  wave  be 
plane  polarized,  and  its  plane  of  polarization  inclined  to  that  of  inci- 
dence. The  vibrations  will  be  resolved  into  their  components,  respec- 
tively parallel  and  perpendicular  to  this  latter   plane ;    and    as    long   as 

sin  9  <  m,  two  components  will  be   reflected  and  two  refracted.      If  — 

be  any  even  multiple  of  i,  in  both  sets  of  components,  the  reflected 
and  intfomitted  resultant  waves  will  be  plane  polarized. 

The  inclinations,  denoted  by  a'  and  at ,  of  the  planes  of  polariza- 
tion of  the  reflected  and  refracted  waves,  respectively,  to  the  plane  of 
•ncidence,  will  be  given  by 

.        V  u 

tan  a  =  -  •  tan  a  ;     tan  cr.   =  —  .  tan  a : 
v  u 

in  which  y,  v\  u  and  u\  are  to  be  found  by  Equations  (597),  (600), 
(598),  and  (607). 

If  a  =  45°  and  sin  9  =  wt,  then  will 

tan  a!  =  —  1,  and  tan  a   =  — . 

m 

At  this  limit,  the  refracted  wave  takes  the  direction  of  the  deviating 
surface.  An  infinitesimal  increment  to  9  will  cause  this  wave  to  be 
reflected  and  make  m  =  —  1,  tan  at  =  —  1,  and  give  to  tan  a!  the 
form  of  indetermination.  But,  retaining  the  limiting  Talue  of  this  fane- 
lion  above,  we  have, 

1  -f  tan  a' .  tan  at  =  1  —  1  =  0 ; 


MECHANICS    OF    MOLECULES.  411 

and  since  the  planes  of  polarization  pass  through  the  same  line,  viz., 
a  normal  to  the  wave  front,  they  will  make  with  one  another  an  angle 
of  90°,  and  the  whole  reflected  wave  will  be  compounded  of  two  equal 
components  polarized  in  planes  at  right  angles  to  each  other.  If  these 
waves  reach  the  molecules   in   their  common  path,  so  as  to  satisfy  the 

condition  that  —  shall   be    an    even  multiple  of  ^,   the   resultant   wave 

will  be  plane  polarized  ;  if  an  odd  multiple,  then  circularly  polarized  ; 
and  if  between  these  limits,  then  elliptically  polarized. 

§  341. — If  the  polarization  be  circular,  then  will  olx  =  ay  =  ay,  be 
equal  to  the  radius  vector  of  the  circular  orbit.  Denote  the  angle 
which  this  radius  makes  with  the  axis  or,  at  any  instant,  by  6 ;  then 
will 

V„  A  -  z 


oLt  .  cos  $  =  £  =  a, .  sin  2  ft 


X 


X 


V  .  t  —  z 
ai  .  sin  6  =  7)  =ay  .  sin  2  n  — ^- . 

Denote  the  time  required  for  the  first  wave  to  describe   Vg .  t  —  z,  by  tu 
that  for  the  second  to  describe  Vy .  t  —  z  by  ty,  and  the  periodic  time 
of  a  molecule   in    both   waves  by  r ;    then,  because    the    wave    velocity 
is   constant,  and   the    wave  length  and  orbit  are  described  in  the  same, 
time, 

V* .  t  -  z  _  t^_        Vv.t-z  =  t, 
A.        ~  r  '  A,  ? 

which,  in  the  above,  give, 

cos  6  =  sin  2  n  .  — , 

T 

sin  6  =  sin  2  it  .  —  ; 

r 

and  making 

t,=  t9±it\ (613) 

in  which  t'  denotes   the   time   the    wave  due   to   vibrations   parallel  to 


412  ELEMENTS    OF    ANALYTICAL    MECHANICS 


• 


one  axis  is  in  advance  of  that  due  to  th>se  parallel  to  the  other;  w$ 
have, 

cos  0  =  sin  2  v    —  ; (614) 


« 


sin  6  =  sin  2 


"    (^±^)  •  '     (615) 


Differentiating,  regarding  —  as  constant,  we  find, 


dd  2n 

-=—  = .  cos  2 

at,       r  .  cos  v 


and,  developing  the  last  factor, 


» 


d0  2tt  4  r         .    _        £,      .    i      t' 

dtr       r  .  cos  6 


cos  2  77  •  —  •  cos  2  rr  •  —  ^  sin  2  77  •  —  •  sin  2  77  —  I ; 


and  making  —  =  |, 


n    d$  2r     .  L  ■  , 

cosd.-r-  =  ^f  — .sin  2  77.— (616) 

a  L  r  r 


Differentiating  (614),  we  find, 


sinfl.— -  «fc _.cos2  7r— (617) 

(It-  T  T 


Squaring,  adding   to  the    square   of   Equation   (616),   and   taking   square 

root, 

dd  277 

17  =  *- <618) 

whence  the  velocity  is  constant. 

The  first  member  of  Equation  (616)  is  the  velocity  in  the  direction 
of  the  axis  y,  and  Equation  (617)  in  the  direction  of  the  axis  as,  and 
these  equations  show  that  the  upper  sign  must  be  taken  in  Equation 
(618)  when  if  is  positive  in  Equation  (613),  and  the  lower  when  t '  is 
negative.  Whence  it  appears,  that  two  waves  plane  polarized  will,  by 
their  simultaneous  action  upon  a  molecule,  cause  it  to  move  uniformly 
in  a  circle,  provided  they  be  of  the  same  length,  and  one  wave  lag, 
as   it    were,  behind    the    other,    bv    a    <"  stance  equal    to    \    of   a    wave 


MECHANICS    OF    MOLECULES.  415 

length  ;  and  the  motion  will  be  from  right  to  left,  or  the  converse,  ac- 
cording to  wave  precedence. 

Two  waves  distinguished  by  these  peculiarities  are  said  to  be  oppo- 
sitely polarized.  The  plane  perpendicular  to  the  wave  front,  and! 
through  that  diameter  of  the  orbit  into  which  the  molecule  would  be 
brought  at  the  same  instant  by  the  separate  action  of  the  two  wave* 
is  called  the  plane  of  crossing. 

§  342.— Let 
( 1 ) ay  cos  Q  =  £  =  a,  sin  2  n  — , 

*                                                                    /           *         t'\ 
(2) a^  sin  0  =  tj  =  a,  sin  I  2  7r  .  —  -\ 1, 

(3)  .     .     .     .     .     .     a4  cos#  =  |  =  ay  sin  |2ip  .  -*-  +  --V 

i 

(4) a,  sin  0  =  7/  =  a,  sin  2  7T  — > 

T 

be  the  displacements  in  two  oppositely  circularly  polarized  waves,  Tlk* 
union  of  (1)  and  (4)  gives  a  resultant  wave  plane  polarized  ;  that  of 
(2)  and  (3)  also    a    wave    plane    polarized,    the    equation    of   the-    path 


being 


s  =  v 


in  the  plane  of  crossing.  It  thus  appears  that  the  union  of  two  circu- 
larly polarized  waves,  polarized  in  opposite  directions,  gives  a  plane- 
polarized  wave,  of  which  the  intensity  is  double  of  either.  Conversely* 
a  wrave  plane  polarized  may  be  resolved  into  two  components  of  equal 
intensity,  circularly  polarized  in  opposite  directions- 

§  343. — Because  the  time  of  describing  the  wave  length  is  equal  to 
the  molecular  periodic  time,  we  have,  denoting  the  velocity  of  wave 
propagation  by  V, 

X  =  Vt, 

whence 

X 


414  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

which,    in    Equation    (618),  gives,    after    multiplying  b}7   tx  and  dividing 
by  2tt, 

<[[*'_^_Yi* (Gl9) 

2tT      ~~      X 

,The  first  member  is  the  arc,  expressed  in  circumferences,  described  by 
the  molecule  while  the  wave  is  moving  through  a  thickness  V .  ta  of 
the  medium.  So  that  a  wave,  compounded  of  many  components  hav 
ing  different  wave  lengths,  but  all  polarized,  on  entering  a  medium, 
may  emerge  with  the  planes  of  polarization  of  its  several  components 
so  twisted  through  different  angles  as  to  diverge  from  a  common  line 
perpendicular  to  the  wave  front.  The  department  of  optics  furnishes 
some  fine  examples  of  this.  A  piece  of  quartz,  of  a  peculiar  kind,  is 
known  to  twist  the  extreme  red  wave  through  an  angle  of  17°  29'  47", 
and  the  extreme  violet,  44°  04'  58",  for  each  millimetre  of  thickness. 

DIFFUSION    AND    DECAY    OF    LIVING    FORCE. 

§  344. — The  living  force  of  any  molecule  whose  mass  is  m  and  ve- 

locitv  v, .  is 

m  v* ; 

and  denoting  by  n  the  number  of  molecules  on  a  superficial  unit  of 
tf.he  wave  front,  the  living  force  on  this  unit  will  be 

n  .  m  . vj  ; 

and  on  the  surface  of  a  spheie  of  which  the  radius  is  rf, 

4  7r .  rf .  n  .  m  v* ; 

nnd  for  another  sphere,  of  which  the  radius  is  r(t,  and  molecular  velo* 

4  n  .  riit .  n  m  vj. 

If  these  spherical  surfaces  occupy  the  same  relative  positions  in  a  di- 
verging wave,  in  any  two  of  its  positions,  their  molecular  living  foioes 
must  be  equal ;  whence,  suppressing  the  common  factors, 

t}  .  m  v  2  =  r  2  m  v  * (620) 


MECHANICS     DF    MOLECULES.  415 

The  molecules  describe  elliptical  c  rbits,  and  under  the  action  of  molec- 
ular forces  directed  to  the  centres  of  these  curves.  The  periodic  time 
will,  therefore,  §  207,  Equation  (286),  be  constant,  however  the  dimen- 
sions of  these  orbits  may  vary ;  and  the  average  velocities  of  the  mole- 
cules will  be  proportional  to  the  lengths  of  their  respective  orbits,  or, 
in  similar  orbits,  to  anv  homologous  dimensions  of  the  same — as  their 
transverse  axes  or  greatest  molecular  displacements.  Denoting  the  latter 
by  c'  and  c"  in  the  two  waves,  then  will 

which,  with  Equation  (620),  gives 

c"  ru  =r.c'rt (621) 

"Whence   it   appears,  that   the  living  force  of  the  molecules  of  any  wave 

* 

varies  inversely  as  the  second,  and  the  greatest  displacement  inversely  as 
the  first  power  of  the  distance  to  which  the  wave  has  been  propagated 
from  its  place  of  primitive  disturbance. 


INTERFERENCE, 


§  345. — Resuming  Equation  (586),  viz., 


-,  +  -h j  -  sin2  2  n  -  =  0  ; 


denote  the  radius  vector  of  the  molecular  orbit  by  p',  and  the  angle  it 
makes  with  the  axis  of  g  by  0',  then  will 


£  =  p' .  cos  0' ;     tj  =  p\  sin  0' ; 
which,  in  the  above,  give 


i .       * 


P  =  — -     "  —  .  sin  2  ft .  - ; 

l/a,,1  cos' 0' +  a/ sin*  0'  T 


and  making 


a/  •  a/y  t> 


v/a,,1 .  cos»  0' +  a,f .  Bin*  0' 


i\Q  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

we  have 

I 
p'  =  c' .  sin  2  rr .  -        (622) 

§ 

In  this  equation,  p'  is  the  actual  displacement  of  the  molecule  from  ita 

t 

place  of  rest,  and  becomes  a  maximum  when  -  is  any  odd  multiple  of 

t 
i.     If,  however,  there  be  added  to  the  arc  2  rr  -,  an    arbitrary  arc  a, 

tin's  latter  may  be  so  taken  as  to  make  the  maximum  or  any  other 
displacement  occur  at  such  time  and  place  as  we  please,  aud,  there- 
fore, to  give  to  the  molecule  any  particular  phase  at  pleasure,  at  the 
time  t.     We  may  write,  then,  generally, 

p'  =  c' .  sin  ^2  rr  .  -  -f  «')  ?      .     .     .         .     (623) 
and  for  a  second  resultant  wave, 

p"  =  c" .  sin  (2  7T  .  -  +  a")  ; (624) 

and  if  these  waves  act  simultaneously  upon  the  same  molecules,  the  re- 
sultant displacement,  denoted  by  p,  will,  §  306,  be  given  by 

p  —  p'  +  p"  =  c'\  sin  (2  n .  -  +  a'\  +  c"  .  sin  (2  n  .  -  +  a"\. 

Developing   the    circular   functions  aid   collecting  the   coefficients  of 
like  factors,  ( 

t'      ......  .  t 

p  =  (cr  cos  a'  +'c"  cos  a") .  sin  2  n-  -f  (c '  sin  a'  +  V  sin  a")  .  cos  2  n  -  ; 

T  7" 

and  making 

c  cos  a  =  c' .  cos  a'  -}  c"  cos  a", . 

(625) 

c  sin  a  =  c'  sin  a'  -j-  e"  sin  a", 


we 


have 


t  t 

o  ss  f .  cos  a  .  sin  2  tt  .  -  -f  c  sin  a  .  cos  2  n .  -  ; 

t  T 


MECHANICS    OF    MOLECULES. 


417 


p  =  c  sin  y2  7T .  — h  a). 


(626) 


Squaring  Equations  (625),  and  adding, 

c*  =  c'2  +  c"2  +  2  c'  c"  cos  (a' 

and  dividing  the  second  by  the  first, 

c  .  sin  a  -f-  c    .  sin  a 


a"),       .     .     .     (627) 


tan  a  = 


c'  cos  a'  +  c"  cos  a 


.// 


(628) 


From  Equation  (626)  we  see  that  the  resultant  wave  is  of  the  same 
length  as  that  of  the  component  waves  to  which  Equations  (623)  and 
(624)  appertain ;  the  length  being  determined  by  the  molecular  periodic 
time  t;  but  the  value  of  a  in  that  equation  differing  from  a'  and  a" 
in  Equations  (623)  and  (624),  shows  that  the  maximum  displacement  of 
a  given  molecule  does  not  take  place  in  the  resultant  wave  at  the  same 
time  as  in  either  of  its  components. 

§  346. — The  maximum  displacement  in  the  resultant  wave  is  given  by 


c  =  vV*  +  c"2  -{-2  c'  c"  .  cos  (a'  -  a")  ;    .     .     .     (629) 

which  will  be  the  greatest  possible  when  a'  —  a"  ==  0,  and  least  pos- 
sible when  a  —  a"  —  180°  ;  the  maximum  in  the  former  case  being 
given  by 


c  =  e'  +  c" 


and  the  minimum,  by 


e  =  c'  -  c". 


In  the  first  case,  Equation  (628), 

(cf  4-  c")  .  sin  a' 

tan  a  =  )—. ~^ =  tan  a  . 

(c   -|-  c  )  .  cos  a' 

Whence  a  =  a   =  a",  and    the    maximum    displacement  will    occur    al 
the  same  place  and  time  in  the  resultant  and  c  mponent  waves. 


418  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

In  the  second  case,  Equation  (628),  if  we  make  a'  =  180°  -\-  a"t 

(cf  —  c")  •  sin  a"  .,  .  , 

tan  a  —  )- Tr{ -  =  tan  a"  =  tan  (a'  —  180°)  =  tan  a  : 

(c  —  c  ) .  cos  a  v  ' 

that  is,  a  will  be  equal  to  one  at  least  of  the  arcs  a'  ancf  a",  and  the 
greatest  displacement  will    occur    at   the    same    time    and    place  in  the 
resultant  wave  as  in  one  of  its  components. 
If  c'  =  c",  then,  Equation  (629), 

c  =  c'^  2\\  -f-  cos  (a'  —  a")]  ; 
and  because 


2a     —  a 


1  +  cos  (a'  —  a")  =  2  cos2 

t  _  _    ft 
c  =  2  c  .  cos — -,      .    .     .     .     .     .     (630) 


and,  Equation  (628), 

sin  a  -f-  sin  a"  a'  +  a"  , 

tana  = - ==  tan ...     .     (631) 

cos  a  +  cos  a  2  v       ' 

If,  while  c'  and  c"  continue  equal,  we  also  have  a'  —  a"  —  180%  then, 

Equation  (630), 

c  =  0. 

Thus  it  appears  that  two  equal  waves  may  reach  the  same  molecules 
in  such  relative  condition  as  to  keep  them  in  their  places  of  rest;  in 
other  words,  two  equal  waves  may  destroy  one  another. 

§  347. — To  ascertain  the  precise    relation   of  two   waves    which    will 
->ause  this  mutual  destruction,  make,  in  Equation  (623), 

a  =  a    rt7r  =  a     ±  — , 

and  that  equation  becomes, 


p  =  c  .sin 


(n      i         t,       2  7T.r\ 

I27T-  +  a"db 1, 

\        r  2r   /' 


2tt l-.:.a"y}     ....     (632) 


MECHANICS    OF    MOLECULES.  419 

which  becomes  identical  with  Equation  (624)   by  making 


c'  =  c", 


anc 

t  =  t  3:  ^-  t.  .  .      .     .     .  (633) 

Now,  the  same  value  for  t,  in  Equations  (623)  and  (624),  will,  tbi 
equal  values  of  the  arbitrary  arcs  a'  and  a",  determine  the  component 
waves  to  give  to  a  molecule  subjected  to  their  simultaneous  action, 
similar  phases  ;  and  a  value  for  t,  in  the  one,  which  differs  from  that 
in  the  other,  by  one-half,  or  any  odd  multiple  of  one-half,  of  the 
molecular  periodic  time,  opposite  phases.  And,  because  the  waves  pro- 
gress by  a  wave  length  during  each  molecular  revolution,  the  above 
result  shows  that,  when  two  waves  meet,  after  having  travelled  over 
routes,  estimated  from  points  at  which  their  molecular  phases  are  simi- 
lar,  and  which  routes  differ  by  half  or  any  odd  multiple  of  half  a 
wave  length,  they  will  destroy  one  another,  provided  the  waves  have  the 
same  length  and  equal  maximum  molecular  displacements.  This  act,  by 
which  one  wave  destroys  another,  is  called  wave  interference. 

The    same    process    of   combination    will    equally  apply   to   three   or 

more  wave  functions  in  which  r  is  the  same  in  all ;  that  is,  wherein  the 

t  t 

wave  lengths  are  the  same;  for,  in  that  case,  sin  2  tt  .  ~    and    cos2  7r..- 

being  common  factors,  after  developing  each  function  in  the  sum,  the 
resultant  displacement  p  becomes, 

p  =  sin  2  77  .  -  .  2  c'  cos  a'  +  cos  2  7r .  —  .  2  c'  sin  a\ 

T  T 

and  assuming 

c  .  cos  a '■  =  2  c'  cos  a', 
c  .  sin  a  =  2  c'  sin  a' ; 

p  =  c  .  sin  (2  7T  -  -f  a),    .     .     (634) 

T 

thus  making  the  resultant  wave  of  the  same  length  as  that  of  either 
of  its  components. 


420  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

But,  if  the  component  waves  be  not  of  equal  lengths,  the  sum  of 
the  corresponding  functions  cannot  reduce  to  the  form  of  Equation 
(634)fl  because  of  the 
absence  of  common 
factors,  arising  from  a  A  -J^ 
change  in  the  valuo 
of  r  from  one  com- 
ponent to  another.     Such  components  can  never  destroy  one  anothei. 

INFLEXION. 

§  348. — Make,    in    Equation  (621),   r"  =  1,    and    that    equation    be- 
comes 

,       c" 
and  this  value  being  substituted  for  c',  in  Equation  (622J,  gives, 


and  making 


c"  .  n 

t 

p' 

=  —  .  sin  2 

7r  .  -; 

r 

r 

t        Vt- 

■r, 

r             X 

» 

we  have,  omitting  all  the  accents, 

c      .           Vt  —  r 
p  =  -  .  sin  2  7r r — , (635) 

Y  A 

which  is  of  the  same  form  as  Equations  (528),  and  in  which  V  is 
the  velocity  of  wave  propagation ;  /,  the  time  of  its  motion  from 
primitive  disturbance ;  A,  the  wave  length  ;  -,  the  maximum  displace- 
ment of  a  molecule  of  which  the  distance  of  the  place  of  rest  from 
the  point  of  primitive  disturbance  is  r;  and  p  the  actual  displacement, 
at,  the  time  t}  of  this  same  molecule.  And  from  which  it  is  apparent 
that  the  displacements  will  always  be  the  same  for  equal  distances, 
Vt —  r,  behind  the  wave  front. 

Every  disturbance  of  a   molecule,  at  one  time,  becomes   a    caiwe  of 


MECHANICS    OF    MOLECULES.  421 

jisturbar.ee  to  another  molecule  at  some  subsequent  time.  All  the 
molecules  in  a  wave  front,  when  they  first  begin  to  move,  become, 
therefore,  centres  of  disturbance  for  every  molecule  in  advance ;  and 
if  the  primitive  disturbance  be  kept  up,  secondary  waves  proceeding 
from  these  centres  will  reach  a  molecule  in  advance  simultaneously, 
and  determine,  §  307,  at  any  instant  f,  its  displacement  2  p. 

Suppose     a     wave,    whose 
centre  of  disturbance  is   (7,  to  t  \  __— -^ss^D 

have  reached  the  position  AB, 
so  remote  from  C  that  a  small 
portion,  A  B,  may  be  regarded 
as    sensibly   plane :     What   is 

the  displacement  of  a  molecule  at  0,  produced  by  the  simultaneous 
action  of  the  secondary  waves  proceeding  from  the  molecules  in  any 
portion,  as  AB,  of  a  section  of  this  wave  front?  Draw  the  normal 
C D  iV,  through  the  middle  of  P  Q  ;  denote  the  variable  distance  D  Q 
by  z,  and  Q  0  by  r.  The  displacement  of  the  molecule  0,  by  the 
secondary  waves  from  the  arc  AB  =  2  6,  will,  Eq.  (635),  be  given  by 

/+6               r+b  cdz                    Vt  —  r 
pdz  —  l         .sin2rr.  = .     .     (636) 
—  b              J  —b      r                             A 

Here  r  and  z  are  variable.  To  eliminate  the  former,  join  0  with  the 
middle  of  AB  by  the  line  D  0,  and  denote  its  length  by  /,  and  the 
angle   Q  D  0,  which  it  makes  with  the  wave  front,  by  0.     Then  wHI 


r  =  Vl*  +  z>—  2/zcos0; 
and  by  Maclaurin's  formula, 

r  =  I  —  cos  B  .  z  +  5^-j-  .  z%  —  <fec (637) 

If  the  greatest  value  of  z  be  small  as  compared  to  /,  we  may  take 

r  =  Z-cos0.z,  (6S8) 

and  regard  the  displacements  of  the  molecule   0,  by  the  partial  waves 

28 


422  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

from  z  to   be   equal.     Whence,  substituting   the    value  of  r,  with   this 
restriction,  in  Equation  (636),  we  have, 

p.dz  =  T.l        sin  —  (Vt  —  I  +  cos0  .  z)  dz% 

and,  performing  the  integration  without  regard  to  limits, 

2 p  = : s  .  cos  -v-  ( V t  —  I  -f  cos  0  .z\ 

r  2  IT  I  COS  0  X     x 

and  between  the  limits  —  b  and  +  b, 

2p= f ^  •  Tcos  V-  (F*— /— 6cos0)— cos-r-  (Vt—l -f 6.  cos (9)  J, 

27r./.cos0L        A  A   v  J 

or, 

cX            .    2  7r.6.cos0      .■             Vt  —  l  tnc,^ 

2p  =r ; j:  •  sin r •  sin  2  tt - ;      ....     (639) 

r  TT  .  I  .  COS  0  A  A 

so  that  the  function  whose  value  gives  the  resultant  displacement,  is  of 
the  same  form  as  that  of  the  function  which  determines  either  of  the 
partial  displacements. 

The  maximum  value  of  the  resultant  displacement  is  given  by 

_  C.X  .     2  7T.6.COS0  /„\,^v 

2  p  = .  sin 1 ;        .     .     .     (640) 

y       n.l.cosd  X  '  x       ' 

and  this  will  become  zero  for  such  values  of  6  as  make  b  .  cos  6  equal 
to  either  of  the  following  values,  viz., 

¥  A,     -j  A,      -%  A,      ^  A,    &C. 

Conceiving  the  fig'ire  to  be  revolved  about  the  normal  C Ny  and  al! 
the  wave  except  the  circular  portion  whose  diameter  is  2  b  =  A  JB,  to 
be  intercepted,  the  space  in  advance  of  the  wave  will,  when  the  above 
values  obtain,  find  itself  divided  by  the  secondary  waves  into  a  series 
of  concentric  cone-like  zones  around  the  normal  CJV,  as  an  axis,  and 
of  which  the  alternate  ones,  beginning  with  that  immediately  about  the 
axis,  will  be  filled  with  molecules  in  motion,  while  the  molecules  in  the 


MECHANICS    OF    MOLECULES.  423 

others  will  be  at  rest.  A  section  in  advance  of  the  primitive  wave 
will  cut  from  these  zones  a  series  of  concentric  circular  rings  distin- 
guished by  the  same  peculiarities. 

But  if  X  be  very  great  as  compared  with  6,  then  will  the  arc 

2  7T  .  b  .  cos  6 


be  so  small  as  to  justify  the  substitution  of  the  arc  for  its  sine  and  for 
the  maximum  value  of  resultant  displacement, 

>_ . .  c  X  2  rr  .  b  .  cos  0       2  c  b 

(sp>'  =  indole x —  =  —  ;-  •  '  (ti4I> 

•  ■ 

and  this  result  being  independent  of  0,  the  conic  zones  cannot  exist, 
and  the  effect  of  the  secondary  waves  will  be  diffused  in  all  directions 
to  Ihe  front.  This  lateral  action  of  secondlirv  waves  proceeding  from 
a  small  portion  of  a  primitive  wave,  is  called  wave  inflection. 

When  6  approaches  nearly  to  90°,  cos  6  will   be   exceedingly  small, 
and  the  arc 

2  n  .  b  .  cos  6 


may  again  be  substituted    for    its  sine  ;    again  Equation  (641)  suits  the 

case,  and  determines  the  maximum  displacement  immediately  about  the 

normal. 

The    maximum    of   the    maxima   displacements  will    occur  when,  in 

Equation  (640), 

2  n  .  b  .  cos  6         . 
sin  . r =  ±  1 ; 

and  which  would  reduce  that  equation  to 

c  X 


(*rt„ 


IT  .  I .  cos  6  ' 


and  as  the  living  forces  are  proportional  to  the  squares  of  the  greatest 
displacements,  we  have 

4c*6J  <rUf 


m  .  v  *  :  m  .  v 


,     •    '"•"„      '    '        p        '   Wt./,C0B,«- 


424  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Whence 

in  which  v  is  the  velocity  of  the  molecule  on  the  normal,  and  t» 
that  at  the  angular  distance  6  from  it.  When  the  waves  are  ven 
short,  as  compared  with  6,  it  is  obvious  that  the  living  force  of  the 
molecules  would  be  sensibly  nothing,  except  immediately  about  the 
normal.  When  the  waves  are  long,  as  compared  with  6,  the  living 
force  will  be  appreciable  far  every  value  of  0,  and,  therefore,  in  every 
direction  in  front  of  the  primitive  wave.  The  importance  of  this 
discussioj    will  tc  apparent  in  the  subjects  of  sound  and  light 


PART    IV. 


APPLICATION   OF   THE    PRECEDING    PRINCIPLES   TO 
SIMPLE    MACHINES,    PUMPS,    ETC. 

§  349  — Any  device  by  which  the  action  of  a  force  may  be  received 
at  one  place  and  transmitted  to  another  is  called  a  Machine. 

Ther^  are  usually  seven  elementary  machines  discussed  in  Me- 
chanics ;  viz.,  the  Cord,  Lever,  Inclined  Plane,  Confined  Fluid,  Pulley, 
Screw,  Wheel  and  Axle,  and  Wedge.  The  Cord,  Lever,  and  Inclined 
Plane  are  called  Simple  Machines;  the  others,  being  combinations  of 
these,  are  called  Compound  Machines. 

§  350. — In  Machines,  as  in  all  other  bodies,  every  action  is  accom- 
panied by  an  equal  and  contrary  reaction.  A  force  which  acts  upon 
a  Machine  to  impress  or  preserve  motion  is  called  a  Power.  A  force 
which  reacts  to  prevent  or  destroy  motion,  is  called  a  Resistance.  The 
Agent  which  is  the  source  of  power,  is,  §  38,  called  a  Motor.  A 
Machine  enables  a  power  to  make  a  resistance  work. 

§  351. — Resuming  Equation  (A),  and  supposing  the  displacement, 
which  in  that  equation  was  wholly  arbitrary,  to  conform  in  every 
respect  to  that  caused  by  the  powers  and  resistances,  we  shall  have 
8  s  =  d  s,  8  being  the  path  described  by  the  elementary  mass  m ; 
and  hence, 

d2  s 
2  PSp  —  2m-  -r-r  .  ds  =  0  ; 

a  r 


but 


whence, 


cP  s  d  s     d2  s 


2  PSp  —  jJm.if(«')=ll.   .....     (643) 


426  ELEMENTS    OF     ANALYTICAL    MECHANICS. 

Denoting    by    Q,   Q ,  &c.  the   resistances,  by  B,  P\  &c.    the   pow 
ers,  5q,  &c.  and    Sp,  &c.  the   projections   of  their    respective    virtual 
velocities  ;     the    first    term,    which    embraces    all     the    forces    except 
inertia  in  action  on  the  machine,  may  be  replaced   by  2PSp  —  2  Qoq> 
and  we   have 

ZPSp  —  SQSq  ==  £2j».rff».    ....     (64J) 
Integrating, 

J%2PSp-f2Q6q  =  $2?nv*+   C; 

and  denoting  by  vt  the  initial  velocity,  and  taking  the  integral  so 
as    to    vanish   when    t  sz  0, 

J  2  PSp  —  J  2  QSq  =  Jlmt;2  -  jlw^..  .  .  (645) 

The  products  P  Sp  and  $  £  q  are  the  elementary  quantities  of 
work  performed  by  a  power  and  a  resistance  respectively,  it 
the  element  of  time  d  t ;  the  product  %mdv2  is  the  elementary 
quantity  of  work  performed  by  the  inertia,  or  one  half  the  incre 
ment  of  living  force  of  the  mass  m  in  this  time.  And  Equation 
(645)  shows  that  in  any  machine,  in  motion,  the  increment  of  the 
half  sum  of  the  living  forces  of  all  its  parts  is  always  equal  to 
the  excess  of  the  work  of  the  powers  or  motors  over  that  of  the 
resistances 

§  352. — If  the   machine   start   from    rest,  Equation  (645)  becomes 
fzPSp—  j*2  Q8q  =  ±2mv2,  •     -     -     -     (646) 

and  as  the  second  member  is  essentially  positive,  the  work  of  the 
motors    must   exceed   that   of   the   resistances   embraced    in  the  term 

flQfiq;   in   other    words,  the   inertia   will    oppose   the     motor    and 

act  as  a  resistance.  When  the  motion  becomes  uniform,  the  second 
member  will  be  constant ;  from  that  instant  inertia  will  cease  to 
act,  and  tha  subsequent  work  of  the  motor  will  be  equal  to  that 
of  the  resistances  as  long  as  this  motion  continues.  If  the  motion 
be  now  retarded,  the  second  member  will  decrease,  the  inertia*  will 
act  with    the   power,  und    this    will    continue   till    the   machine   com*** 


APPLICATIONS.  427 

to  rest,  and  the  excess  <f  work  of  the  Resistance  during  retardation 
will  be  exactly  equal  to  that  of  the  Power  during  acceleration. 
Generally,  then,  when  a  machine  is  at  rest  or  is  moving  uniformly, 
inertia  does  not  act ;  when  the  motion  is  variable,  it  does,  and 
opposes  or  aids  the  motor  according  as  the  motion  is  accelerated 
or  retarded. 

§353. — The  essential  parts  of  every  machine  are  those  which 
receive  erectly  the  action  of  the  motor,  those  which  act  directly 
upon  the  body  to  be  moved  or  transformed,  and  those  which  serve 
to  transmit  the  action.  The  arrangement  of  the  latter  is  often  a 
source  of  resistance,  arising  from  Friction,  Adhesion,  Stiffness  of 
Cordage,    &c,    whose    work    enters    largely    into    the    general    term 

fzQSq. 

FRICTION. 

§  354. — When  two  bodies  are  pressed  together,  experience  shows 
that  a  certain  effort  is  always  required  to  cause  one  to  roll  or  slide 
along  the  other.  This  arises  almost  entirely  from  the  inequalities  in 
the  surfaces  of  contact  interlocking  with  each  other,  thus  rendering 
it  necessary,  when  motion  takes  place,  either  to  break  them  off,  com- 
press them,  or  force  the  bodies  to  separate  far  enough  to  allow  them 
to  pass  each  other.  This  cause  of  resistance  to  motion  is  called  fric- 
tion,  of  which  we  distinguish  two  kinds,  according  as  it  accompanies 
a  sliding  or  rolling  motion.  The  first  is  denominated  sliding,  and 
the  second  rolling  friction.  They  are  governed  by  the  same  laws ; 
the  former  is  much  greater  in  amount  than  the  latter  under  given 
circumstances,  and  being  of  more  importance  in  machines,  will  prin- 
cipally occupy   our  attention. 

The  intensity  of  friction,  in  any  given  case,  is  measured  by  the 
force  exerted  in  the  direction  of  the  surface  of  contact,  which  will 
place  the  bodies  in  a  condition  to  resist,  during  a  change  of  state, 
in  respect  to  motion  or  rest,  only  by  their  inertia. 

§355. — The  friction  between  two  bodies  maybe  measured  directly 
by   means  of  the  spring  balance.      For  this   purpose,  let  the  surface 


428 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


C  D  of  one   of  the   bodies  M  be   made   perfectly  level,  so   that    th* 

other  bodv  M\  when  laid 

upon   it,  may  press    with 

its  entire  weight.  To  some 

point,  as  F,  of  the   body 

M\  attach  a  cord  with  a 

spring     balance     in      the 

manner  indicated  in  the  figure,  and    apply  to  the  latter  a  force  F  of 

such  intensity  as  to  produce  in  the  body  M  a  uniform  motion.      The 

motion    being    uniform,  the  accelerating  and  retarding  forces  must  be 

equal    and    contrary;    that  is  to  say,  the  friction    must  be   equal  and 

contrary  to    the  force  F,  of  which    the    intensity  is    indicated   by  the 

balance. 

The  experiments  on  friction  which  seem  most  entitled  to  conf 
dence  are  those  performed  at  Metz  by  M.  Morin,  under  the  ordei 
of  the  French  government,  in  the  years  1831,  1832,  and  1833.  They 
were  made  by  the  aid  of  a  contrivance,  first  suggested  by  M.  Pon- 
celet,  which  is  ono  of  the  most  beautiful  and  valuable  contributions 
that  theory  has  ever  made  to  practical  mechanics.  Its  details  are 
given  in  a  work  by  M.  Morin,  entitled  "  Nouvelles  Experiences  sur  le 
FrotternenC     Paris,  1833. 

The  following  conclusions  have  been  drawn  from  these  experi- 
ments, viz. : 

The  friction  of  two  surfaces  which  have  been  for  a  considerable 
time  in  contact  and  at  rest  is  not  only  different  in  amount,  but  also 
in  nature,  from  the  friction  of  surfaces  in  continuous  motion  ;  espe- 
cially in  this,  that  the  friction  of  quiescence  is  subjected  to  causes  of 
variation  and  uncertainty  from  which  the  friction  during  motion  is 
exempt.  This  variation  does  not  appear  to  depend  upon  the  extent 
of  the  surface  of  contact;  for,  with  different  pressures,  the  ratio  of 
the  friction  to  the  pressure  varied  greatly,  although  the  surfaces  of 
contact  were  the  same. 

The  slightest  jar  or  shock,  producing  the  most  imperceptible 
movement  of  the  surfaces  of  contact,  causes  the  friction  of  quies- 
cence to  pass  to  that  which  accompanies  motion.  As  every  machine 
may  be  regarded  as  being  subject  to  slight  shocks,  producing  imper 


APPLICATIONS  429 

ceptible  motions  in  the  surfaces  of  contact,  the  kind  o:'  friction  to  be 
employed  in  all  questions  of  equilibrium,  as  well  as  of  motions  of 
machines,  should  obviously  be  this  last  mentioned,  or  that  which 
accompanies  continuous  motion. 

The  laws  of  friction  which  accompanies  continue  us  motion  are 
remarkably  uniform  and  definite.     These  laws  are : 

1st.  Friction  accompanying  continuous  motion  of  two  surfaces, 
between  which  no  unguent  is  interposed,  bears  a  constant  proportion 
to  the  force  by  which  those  surfaces  are  pressed  together,  whatever 
be  the  intensity  of  the  force. 

2d.  Friction  is  wholly  independent  of  the  extent  of  the  surfaces  in 
contact. 

3d.  Where  unguents  are  interposed,  a  distinction  is  to  be  'made 
between  the  case  in  which  the  surfaces  are  simply  unctuous  and  in 
intimate  contact  with  each  other,  and  that  in  which  the  surfaces  are 
wholly  separated  from  one  another  by  an  interposed  stratum  of  the 
unguent.  The  friction  in  these  two  cases  is  not  the  same  in  amount 
under  the  same  pressure,  although  the  law  of  the  independence  of 
extent  of  surface  obtains  in  each.  When  the  pressure  is  increased 
sufficiently  to  press  out  the  unguent  so  as  to  bring  the  unctuous  sur- 
faces in  contact,  the  latter  of  these  cases  passes  into  the  first;  and 
this  fact  may  give  rise  to  an  apparent  exception  to  the  law  of  the 
independence  of  the  extent  of  surface,  since  a  diminution  of  the  sur- 
face of  contact  may  so  concentrate  a  given  pressure  as  to  remove  the 
unguent  from  between  the  surfaces.  The  exception  is,  however,  but 
apparent,  and  occurs  at  the  passage  from  one  of  the  cases  above- 
named  to  the  other.  To  this  extent,  the  law  of  independence  of  the 
extent  of  surface  is,  therefore,  to  be  received  with  restriction. 

There  are,  then,  three,  conditions  in  respect  to  friction,  under 
which  the  surfaces  of  bodies  in  contact  may  be  considered  to  exist, 
viz.:  1st,  that  in  which  no  unguent  is  present;  2d,  that  in  which 
the  surfaces  are  simply  unctuous;  3d,  that  in  which  there  is  an 
interposed  stratum  of  the  unguent.  Throughout  each  of  these  states 
the  friction  which  accompanies  motion  is  always  proportional  to  the 
pressure,  but  for  the  same  pressure  in  each,  very  different  in 
amount. 


£30  ELEMENTS     OF     ANALYTICAL     MECHANICS 

4th.  The   friction    which   accompanies   motion    is   always   independ 
ent  of  the  velocity  with  which  the   bodies   move;    and    this,   whether 
the     surfaces    be   without    unguents    or     lubricated   with    water,    oils, 
grease,  glutinous  liquids,   syrups,  pitch,   &c,   &c. 

The  variety  of  the  circumstances  under  which  those  laws  obtain, 
and  the  accuracy  with  which  the  phenomena  of  motion  accord  with 
them,  may  be  inferred  from  a  single  example  taken  from  the  first 
set  of  Morin's  experiments  upon  the  friction  of  surfaces  of  oak, 
whose  fibres  were  parallel  to  the  direction  of  the  motion.  The  sur- 
faces of  contact  were  made  to  vary  in  extent  from  1  to  84 ;  the 
forces  which  pressed  them  together  from  88  to  2205  pounds ;  and 
the  velocities  from  the  slowest  perceptible  motion  to  9,8  feet  a 
second,  causing  them  to  be  at  one  time  accelerated,  at  another 
uniform,  and  at  another  retarded ;  yet,  throughout  all  this  wide 
rano-e  of  variation,  in  no  instance  did  the  ratio  of  the  friction  to 
the  pressure  differ  from  its  mean  value  of  0,478  by  more  than  ^ 
of  this  same  fraction. 

Denote  the    constant    ratio    of   the    entire    friction  F,  to  the  normal 
'pressure  P,  by  /;  then   will    the   first  law  of  friction    be    expressed    by 
the  following  equation, 

y=A m> 

whence, 

v«;:  This   constant  ratio  /  is   called   the  co-ejicimt   of  friction,   because, 
'when    multiplied   by    the   total    normal    pressure,    the   product   gives 

the    entire  friction. 
,  Assuming   the    first   law  of  fric- 

tion,  the  co-efficient  of  friction  may 

easily  be  obtained  by  means  of  the 

inclined  plane.     Let   W  denote  the 

weight   of    any   body    placed   upon 

the   inclined    plane    A  B.      Resolve 

this,  weight  G  G'   into   two   compo- 
nents,   one    GM   perpendicular    to 

the   plane,  and  the  other   G  JV  par 


APPLICATIONS.  431 

allel    to   it.      Because  the   angles    G'  G  M  and   BAC  are   equal,  the 
.  first   of  these   comporents  will  be 

GM  =  W.cosA, 
and    the   second, 

GN  =  W.sinA, 

in  which  A    denotes    the   ansjle  B  A  C. 

The  first  of  these  components  determines  the  total  pressure  ujk>d 
the    plane,  and    the   friction  due    to    this   pressure  will    be 

F  =  /.  W  cos  A. 

The  second  component  urges  the  body  to  move  down  the  plane, 
ff  the  inclination  of  the  plane  be  gradually  increased  till  the  body 
move  with  uniform  motion,  the  total  friction  and  this  component 
must   be    equal    and    opposed ;    hence, 

/.  W .  cos  A  =  W .  sin  A  ; 

vhence, 

.       sin  A  ■ 

f  — =  tan  A. 

cos  A 

We,  therefore,  conclude,  that  the  unit  or  co-efficient  of  friction 
between  any  two  surfaces,  is  equal  to  the  tangent  of  the  angle 
which  one  of  the  surfaces  must  make  with  the  horizon  in  ordei 
that  the  other  may  slide  over  it  with  a  uniform  motion,  the  body 
to  which  the  moving  surface  belongs  being  acted  upon  by  its  own 
weight  alone.  This  angle  is  called  the  angle  of  friction  or  limiting 
angle    of  resistance. 

The  values  of  the  unit  of  friction  and  of  the  limiting  angles  for 
many  of  the  various  substances  employed  in  the  art  of  construction, 
are  given  in  Tables  VI,  VII    and  VIII. 

The  distinction  between  the  friction  of  surfaces  to  which  no  un 
guent  is  applied,  those  which  are  merely  unctuous,  and  those  between 
which  a  uniform  stratum  of  the  unguent  is  interposed,  appears  first 
to  have  been  remarked  by  M.  Morin  ;  it  has  suggested  to  him 
what  appears  to  be  the  true  explanation  of  the  Jifference  between 
his    results    and    those    of  Coulomb.      He    conceive*,  that    in    the   ex- 


432  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

periments  of  this  celebrated  Engineer,  the  requisite  precautions  had 
not  been  taken  to  exclude  unguents  from  the  surfaces  of  contact. 
The  slightest  unctuosity,  such  as  might  present  itself  accidentally, 
unless  expressly  guarded  against — such,  for  instance  as  might  have 
been  left  by  the  nands  of  the  workman  who  hn>-)  given  the  last 
polish  to  the  surfaces  of  contact — is  sufficient  materially  to  affect 
the   co-efficient  of  friction. 

Thus,  for  instance,  surfaces  of  oak  having  been  rubbed  with  hard 
dry  soap,  and  then  thoroughly  wiped,  so  as  to  show  no  traces 
whatever  of  the  unguent,  were  found  by  its  presence  to  have  lost 
|ds  of  their  friction,  the  co-efficient  having  passed  from  0,478 
to  0,164 

This  effect  of  the  unguent  upon  the  friction  of  the  surfaces  may 
be  traced  to  the  fact,  that  their  motion  upon  one  another  without 
unguents  was  always  found  to  be  attended  by  a  wearing  of  both  the 
surfaces ;  small  particles  of  a  dark  color  continually  separated  from 
them,  which  it  was  found  from  time  to  time  necessary  to  remove, 
ind  which  manifestly  influenced  the  friction  :  now,  with  the  presence, 
of  an  unguent  the  formation  of  these  particles,  and  the  consequent 
wear  of  the  surfaces,  completely  ceased.  Instead  of  a  new  surface 
of  contact  being  continually  presented  by  the  wear,  the  same  surface 
remained,  receiving    by  the  motion  continually  a  more  perfect  polish. 

A  comparison  of  the  results  enumerated  in  Table  VIII,  leads  to 
the  following  remarkable  conclusion,  easily  fixing  itself  in  the  memory, 
that  with  the  unguents,  hogs'  lard  and  olive  oil  interposed  in  a  con- 
tinuous stratum  between  them,  surfaces  of  wood  on  metal,  wood  on 
wood,  metal  on  wood,  and  metal  on  metal,  when  in  motion,  have  all 
of  them  very  nearly  the  same  co-efficient  of  friction,  the  value  of  thai 
co-efficient  being  in  all  cases  included  between  0,07  and  0,08,  and  the 
limiting   angle   of  resistance  therefore  between  4°  and  4°  35'. 

For  the  unguent  tallow  the  co-efficient  is  the  same  as  the  above  in 
every  case,  except  in  that  of  metals  upon  metals;  this  unguent  seems 
less  suited  to  metallic  surfaces  than  the  others,  and  gives  for  the 
mean  value  of  its  coefficient  0,10,  and  for  its  limiting  angle  of  re- 
sistance 5°  43 


APPLICATIONS. 


433 


356. — Besides  friction,  there  is  another  cause  of  resistance  to  the 
motion  of  bodies  when  moving  over  one  another.  The  same  forces 
which  hold  the  elements  of  bodies  together,  also  tend  to  keep  the 
bodies  themselves  together,  when  brought  into  sensible  contact.  The 
effort  by  which  two  bodies  are  thus  united,  is  called  the  force  of 
Adhesion. 

Familiar  illustrations  of  the  existence  of  this  force  are  furnished 
by  the  pertinacity  with  which  sealing-wax,  wafers,  ink,  chalk  and 
black-lead  cleave  to  paper,  dust  to  articles  of  dress,  paint  to  the 
surface  of  wood,  whitewash  to  the  walls  of  buildings,  and  the  like. 
'  The  intensity  of  this  force,  arising  as  it  does  from  the  affinity 
of  the  elements  of  matter  for  each  other,  must  vary  with  the  num- 
ber of  attracting  elements,  and  therefore  with  the  extent  of  the  sur- 
face of  contact. 

This  law  is  best  verified,  and  the  actual  amount  of  adhesion  be- 
tween different  substances  determined,  by  means 
of  a  delicate  spring-balance.  For  this  purpose, 
the  surfaces  of  solids  are  reduced  to  polished 
planes,  and  pressed  together  to  exclude  the  air, 
and  the  efforts  necessary  to  separate  them  noted 
by  means  of  this  instrument.  The  experiment 
being  often  repeated  with  the  same  substances, 
laving  different  extent  of  surfaces  in  contact,  it 
is  found  that  the  effort  necessary  to  produce 
the  separation  divided  by  the  area  of  the  surface 
gives  a  constant  ratio.  Thus,  let  S  denote  the 
area  of  the  surfaces  of  contact  expressed  in  square 
feet,  square  inches,  or  any  other  superficial  unit; 
A  the  effort  required  to  separate  them,  and  a 
the    constant   ratio  in   question,  then  will 


A_ 


a. 


or, 


A  =  a  .  S. 


The    constant   a   is   called    the   unit   or  co-efficient  of  adhesion,  and  otv 


4:34 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


viously  expresses  the  value  of  adhesion  on  each  unit  of  surface,  for 
making 

S=l, 

we , have 

A  =  a. 

'  To  find  the  adhesion  between  solids  and  liquids,  suspend  the  solid 
from  the  balance,  with  its  polished  surface  downward  and  in  a  hori* 
zontal  position  ;  note  the  weight  of  the  solid, 
then  bring  it  in  contact  with  the  horizontal 
surface  of  the  fluid  and  note  the  indication  of 
the  balance  when  the  separation  takes  place, 
on  drawing  the  balance  up ;  the  difference  be- 
tween this  indication  and  that  of  the  weight 
will  give  the  adhesion ;  and  this  divided  by 
the  extent  of  surface,  will  give,  as  before,  the 
co-efficient  a.  But  in  this  experiment  two 
opposite  conditions  must  be  carefully  noted, 
else  the  cohesion  of  the  elements  of  the  liquid 
for  each  other  may  be  mistaken  for  the  adhe- 
sion of  the  solid  for  the  fluid.  If  the  solid 
on  being  removed  take  with  it  a  layer  of  the 
fluid ;    in    other    words,    if    the   solid    has   been 

wet  by  the  fluid,  then  the  attraction  of  the  elements  of  the  solid 
for  those  of  the  liquid  is  stronger  than  that  of  the  elements  of  the 
liquid  for  each  other,  and  a  will  be  the  unit  of  adhesion  of  two 
surfaces  of  the  fluid.  If,  on  the  contrary,  the  solid  on  leaving  the 
fluid  be  perfectly  dry,  the  elements  of  the  fluid  will  attract  each 
other  more  powerfully  than  they  will  those  of  the  solid,  and  a  will 
denote   the   unit   of  adhesion  of  the   solid   for   the   liquid. 

It  is  easy  to  multiply  instances  of  this  diversity  in  the  action  of 
solids  and  fluids  upon  each  other.  A  drop  of  water  or  spirits  of 
wine,  placed  upon  a  wooden  table  or  piece  of  glass,  loses  its  globu- 
lar form  and  spreads  itself  over  the  surface  of  the  solid ;  a  drop  of 
mercury  will  not  do  so.  Immerse  the  finger  in  water,  it  becomes 
wet ;    in   quicksilver,  it   remains   dry.     A  tallow  candle,  or  a  feather 


x 


A  PPLIC  AT10NS. 


435 


from  any  species  of  water-fowl,  remains  dry  the  lgh  dipped  in  water. 
Gold,  silver,  tin,  lead,  &c,  become  moist  on  being  immersed  in 
quicksilver,  but  iron  and  platinum  do  not.  Quicksilver  when  poured 
into  a  gauze  bag  will  not  run' through;  water  will:  place  the  gauze 
containing  the  quicksilver  in  contact  with  water,  and  the  metal  will 
also  flov    through. 

It  is  difficult  to  ascertain  the  precise  value  of  the  force  of  ad  he 
sion  between  the  rubbing  surfaces  of  machinery,  apart  from  that  of 
friction.  But  this  is  attended  with  little  practical  inconvenience,  as 
long  as  a  machine  is  in  motion.  •  The  experiments  of  which  the 
results  are  given  in  Tables  VI,  VII  and  VIII,  and  which  are  applicable 
to  machinery,  were  made  under  considerable  pressures,  such  as  those 
with  which  the  parts  of  the  larger  machines  are  accustomed  to  move 
upon  one  another.  Under  such  pressures,  the  adhesion  of  unguents 
to  the  surfaces  of  contact,  and  the  opposition  to  motion  presented 
by  their  viscosity,  are  causes  whose  influence  may  be  safely  disre 
garded  as  compared  with  that  of  friction.  In  the  cases  of  lighter 
machinery,  however,  such  as  watches,  clocks,  and  the  like,  these 
considerations  rise   into   importance,  and   cannot   be   neglected. 

STIFFNESS    OF    CORDAGE. 


§  357. — Conceive     a    wheel     turning 
freely  about    an    axle    or    trunnion,  and 
having  in  its  circumference  a  groove  to 
receive  a  cord   or   rope.     A  weight    W, 
being   suspended    from    one   end  of  the 
rope,  while  a  force  F,  is  applied  to  the 
other    extremity    to    draw     it    up,    the 
latter   will    experience    a    resistance   in 
consequence  of  the  rigidity  of  the  rope, 
which   opposes    every  effort   to    bend  it 
around  the  wheel.     This  resistance  must, 
of  necessity,  consume  a  portion  of   the 
work  of  the  force  F.     The  measure  of 
the    resistance    due    to    the    rigidity  of   cordage    has    been    made    the 


436  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

subject  of  experiment  by  Coulomb ;  and,  according  to  him,  it 
results  that  for  the  same  cord  and  same  wheel,  this  measure  is 
composed  of  two  parts,  of  which  one  remains  constant,  while  the 
other  varies  with  the  weight  W,  and  is  directly  proportional  to  it ; 
so  that,  designating  the  constant  part  by  K,  and  the  ratio  of  the 
variable  part  to  the  weight  W  by  /,  the  measure  will  be  given  by 
the  expression 

K+  I.  W; 

in  which  K  represents  the  stiffness  arising  from  the  natural  torsion 
or  tension  of  the  threads,  and  /  the  stiffness  of  the  same  cord  due  to 
a  tension  resulting  from  one  unit  of  weight;  for,  making  W  ==  1,  the 
above  becomes 

K  +  /. 

Coulomb  also  found  that  on  changing  the  wheel,  the  stiffness  varied 
in  the  inverse  ratio  of  its  diameter ;   so  that  if 

K+  I.W 

be  the  measure  of  the  stiffness  for  a  wheel  of  one  foot  diameter,  then 

will 

K  +  I.W 
2E 

be  the  measure  when  the  wheel  has  a  diameter  of  2  R.  A  table 
giving  the  values  of  K  and  I  for  all  ropes  and  cords  employed  in 
practice,  when  wound  around  a  wheel  of  one  foot  diameter,  and  sub- 
jected to  a  tension  arising  from  a  unit  of  weight,  would,  therefore, 
enable  us  to  find  the  stiffness  answering  to  any  other  wheel  and 
weight  whatever. 

But  as  it  would  be  impossible  to  anticipate  all  the  different  sizes 
of  ropes  used  under  the  various  circumstances  of  practice,  Couloml) 
also  ascertained  the  law  which  connects  the  stiffness  with  the  diame- 
ter of  the  cross-section  of  the  rope.  To  express  this  law  in  all  cases, 
he  found  it  necessary  to  distinguish,  1st,  new  white  rope,  either  drv 
*»r  moist  j  2d,  white  ropes  partly  worn,  either  dry  or  moist ;  3d,  tarred 
ropes  ;  4th,  packthread.  The  stiffness  of  the  first  class  he  found  nearly 
proportional  to  the  square  of  the  diameter  of  the  cross-section ;    that 


APPLICATIONS.  437 

of  the  second,  to  the  square  root  of  the  cube  of  this  diameter,  nearly ; 
that  of  the  third,  to  the  number  of  yarns  in  the  rope ;  and  that  of 
the  fourth,  to  the  diameter  of  the  cross-section  So  that,  if  &  denote 
the  resistance  due  to  the  stiffness  of  any  given  rope;  d  the  ratio  of 
its  diameter  to  that  of  the  table;  and  n  the  ratio  of  the  numbei  of 
yarns  in  any  tarred  rope  to  that  of  the  table,   we  shall  have  for 

AVjo  white  rope,  dry  or  moist. 
Half  worn  white  rope,  dry  or  moist. 
Tarred  rope. 

K+  I.W 

Packthread. 

b  =  a  -  — — •     (651) 

li    -ti 

For  packthread,  it  will  always  be  sufficient  to  use  the  tabular 
values  given,  corresponding  to  the  least  tabular  diameters,  and  substi- 
tute them  in  Equation  (651).  An  example  or  two  will  be  sufficient 
to  illustrate  the  use  of  these  tables. 

Example  1st.  Required  the  resistance  due  to  the  stiffness  of  a  new 
dry  white  rope,  whose  diameter  is  1,18  inches,  when  loaded  with 
a  weight  of  882  pounds,  and  wound  about  a  wheel  1,64  feet  in 
diameter. 

Seek  in  No.  1,  Table  X,  the  diameter  nearest  that  of  the  given 
rope ;    it  is  0,79  ;    hence, 

''  =  S  =  1'5nearly: 

and  from   the  table  at   the  side, 

d?  =  2,25. 
From  No.   1,  opposite  0,79,  we  find 

K  =  1,6097, 

/  =  0,03195; 
27 


438  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

ft. 

which,   together    with    the    weight    W  =  882   lbs.,    and   2  R  =  1,64, 

substituted  in  Equation   (648),  give 

lb.  lb. 

8  =  2,25  -1'6097  +  Off"  *_ggg  =  4$17, 

which  is  the  true  resistance  due  to  the  stiffness  of  the  rope  in 
question. 

Example  2d.  What  is  the  resistance  due  to  the  stiffness  of  a 
white  rope,  half  worn  and  moistened  with  water,  having  a  diam- 
eter equal  to  1.97  inches,  wound  about  a  wheel  0,82  of  a  foot  in 
diameter,  and   loaded  with  a  weight  of  2205  pounds? 

The  tabular  diameter  in  No.  4,  Table  X,  next  less  than  1,97, 
is  1,57,  and  hence, 

d  =  -~  =  1,3  nearly; 

the   square  root  of  the  cube  of  which  is,  by  the  table  at  the  side, 

di  =  1,482. 
In  No.  4  we  find,  opposite   1,57, 

K  =  6,4324,  . 

/    =  0,06387; 

ft. 
which    values,    together    with    W  =  2205   lbs.,    and   2  R  =  0,82,   in 

Equation  (649),  give 

lbs .  lbs. 

s  =  M82  x  6'4324  +  %gg  x  2205  --=  aeffog, 

0,0^ 

which  is  *,he  required  resistance. 

Example  Sd.  What  is  the  resistance  due  to  the  stiffness  of  a 
tarred  rope  of  22  yarns,  when  subjected  to  the  action  of  a  weight 
equal  to  4212  pounds,  and  wound  about  a  wheel  1,3  feet  diameter, 
the  weight  of  one  running  foot  of  the  rope  being  about  0,6  of  a 
pound  I 

By  referring  to  No.  5,  Table  X,  we  find  the  tabular  number  of 
yarns  next  less  than  22  to  be  15,  and  hence, 

22 
n  =  —  =  1,466  nearly. 
15  J 


APPLICATIONS 


439 


In  the  same   table,  opposite   15,   we  find 

K  =  0,7664, 
/    =  0,019879; 


ft 


which,  together   with    W  =  4212,  and  2  Ji  =  1,3,  in    Equation  (G50), 
give 

S  =  ,,40«  °17664  +  W/fTC  ><  4al2  =  oft* 

l,o 

Example  4tk.  Required  the  resistance  due  to  the  stiffness  of  a 
new  white  packthread,  whose  diameter  is  0,196  inches,  when  moist- 
ened or  wet  with  water,  wound  about  a  wheel  0,5  of  a  foot  in 
diameter,   and  loaded   with  a  weight  of  275  pounds. 

The    lowest    tabular   diameter    is  0,39  of  an    inch,  and    hence 

0.196 


0,390 


=  0,5  nearly. 


In    No.  2,  Table    X,  we  find,  opposite  0,39, 

K  =  0,8048, 
I  =  0,00798  ; 

which,  with    W  =  275,  and    2  R  —  0,5,   we  find,  after   substituting  in 
Equation  (651), 

8  =  05  0,8048  +  0,00708  X^75  =    ^         . 

0,5 

§  358. — The  resistance  just  found 
is  expressed  in  pounds,  and  is  the 
amount  of  wreight  which  would  be 
necessary  to  bend  any  given  rope 
around  a  vertical  wheel,  so  that 
the  portion  A  E,  between  the  first 
point  of  contact  ^4,-and  the  point 
E,  where  the  rope  is  attached  to 
the  weight,  shall  be  perfectly  straight. 
The  entire  process  of  bending  takes 
place  at  this  first  or  tangential 
noint    A  ;    for,  if  motion    be   com- 


440         ELEMENTS     OF     ANALYTICAL     MECHANICS. 

inunieated  to  the  wheel  in  the  direction  indicated  by  the  airow 
head,  the  rope,  supposed  not  to  slide,  will,  at  this  point,  take  ana 
retain  the  constant  curvature  of  the  wheel,  till  it  passes  from  the 
latter  on  the  side  of  the  power  F,  When,  therefore,  by  the  motion 
of  the  wheel,  the  point  m  of  the  rope,  now  at  the  tangential  point, 
passes  to  m',  the  working  point  of  the  force  £  will  have  described 
in  its  own  direction  the  distance  A  D.  Denoting  the  arc  described 
by  a  point  at  the  unit's  distance  from  the  centre  of  the  wheel 
by  84 ,  and   the    radius  of  the    wheel    by   R,   we    shall    have 

AD  m  Rsd; 

and  representing   the   quantity  of  work  of  the  force  S  by  L,  we  get 

L  =  S.Bs,; 

replacing  S  by  its  value  in  Equations  (648)  to  (651), 

r       „       .    K+  I.  W 
L  =  Rsrdt — (652) 

3 

in  which  d4  represents  the  quantity  d2,  rf*,  n,  or  d,  in  Equations  (648) 
to  (651),  according    to   the   nature  of  the    rope. 

Example. — Taking  the  2d  example  of  §357,  and  supposing  a  por- 
tion of  the  rope,  equal  to  20  feet  in  length,  V)  have  been  brought 
•n    contact  with    the    wheel    after    the    motion    begins,   we  shall  have 

L  =  20  X  266,109  =  5322,18    units  of  work; 

that  is,  the  quantity  of  work  consumed  by  the  resistance  due  to 
the  stiffness  of  the  rope,  while  the  latter  is  moving  over  a  distance 
of  20  feet,  would  be  sufficient  to  raise  a  weight  of  5322,18  pounds 
through   a  vertical    height  of   one  foot. 


FRICTION    ON   PIVOTS,    AND   TRUNNIONS. 

§  359. — All  rotating  pieces,  such  as  wheels  supported  upon  other 
pieces,  give  rise  by  their  motion  to  friction.  This  is  an  important 
element,  in  all  computations  relating  to  the  performance  of  machinery. 
It    seems    tc    be    different    according    as    the    rotating    pieces  are    kept 


APPLICATIONS. 


Ul 


in  place  by  trunnions  or  by 
pivots.  By  trunnions  are  meant 
cylindrical  projections  a  a  from 
the  ends  of  the  arbor  A  B  of  a 
wheel.  The  trunnions  rest  on  the 
concave  surfaces  of  cylindrical 
boxes  CD,  with  which  they  usu- 
ally have  a  small  surface  of 
contact  m,  the  linear  elements 
of  both  being  parallel.  Pivots 
are  shaped  like  the  trunnions, 
but  support  the  weight  of  the 
wheel  and  its  arbor  upon  their 
circular  end,  which  rests  against 
the  bottom  of  cylindrical  sock- 
ets FGHL 


w 


PIVOTS. 

Let  N  denote  the  force,  in  the  direction  of  the  axis,  by  whict 
the  pivot  is  pressed  against  the 
bottom  of  the  socket.  This  force 
may  be  regarded  as  passing 
through  the  centre  of  the  cir- 
cular end  of  the  pivot,  and  as 
the  resultant  of  the  partial  pres- 
sures exerted  upon  all  the  ele- 
mentary surfaces  of  which  this 
circle  is  composed.  Denote  by 
A  the  area  of  the  entire  circle, 
then  will  the  pressure    sustained 

by  each  unit  of  surface  be 

N 
A  ' 

and    the   pressure  on  any  small  portion  of  the  surface  denoted  by  a, 

will  obviously  be 

a.N 


442 


ELEMENTS  OF  ANALYTICAL  MECHANICS, 


and  the  friction  on  th<.  same  will  be 

f.a.N 


This  friction  may  be  regarded  as  applied  to  the  centre  of  the  ele- 
nentary  surface  a ;  it  is  opposed  to  the  motion,  and  the  direction  of 
its  action  is  tangent  to  the  circle  described  by  the  centre  of  the 
element.  Denote  the  radius  of  this  circle  by  *,  then  will  the  mo- 
ment of  the  friction  be 

Now.  if  t  denote  the  length  of  any  variable  portion  of  the  circumfer 
enee  at  the  unit's  distance  from  the  centre   (7,  then  will 


also, 


a  =  x  .  d  s  .  d x\ 
A  =  «  E2; 


which  substituted  above  give 


f.N 


x2  .  dx  .  d  s 
«  .R2      ' 


end  by  integration, 


f.N 


x2  d  x    I       d  s 


«  R2 


=  /-Ar-j  A; 


(653) 


whence  we  conclude,  that,  in  the  fric- 
tion of  a  pivot,  we  may  regard  the 
whole  friction  due  to  the  pressure  as 
acting  in  a  single  point,  and  at  a  dis- 
tance from  the  centre  of  motion  equal 
to  two-thirds  of  the  radius  of  the  base 
of  the  pivot.  This  distance  is  called' 
the  wean  lever  of  friction. 

§  360. — If  the  extremity  of  the  pivot, 
instead  of  rubbing  upon  an  entire  circle, 
is  only  in  contact  with  a  ring  or  sur- 
face comprised    between   two   concentric 


APPLICATIONS.  443 

circles,  as   when    the    irbor   of  a  wheel    is   urged   in    the  direction  of 
its  length  by  the  force  N  against  a  shoulder  d  c  b  a  ;    then  will 

A  =  «  (R2  -  It'2)  ; 

and  the  integration  will  give 


pR  /»2  t 

/     x2  dx   I       d 


_  _  2  f.  y.  R*  ~  ^  . 
J  «  (R2  -  R'2)  f/  R2  —  R'2  ' 

in    which   R    denotes    the    radius    of    the    larger,  and  R'  that    of  the 
smaller  circle. 

Finally,  denote  by  I  the  breadth  of  the  ring,  that  is,  the  dis- 
tance Af  A\  by  r,  its  mean  radius  or  distance  from  C  to  a  point 
half  way  between  A'  and  A,  and  we  shall  have 

R'  =  r-il; 
substituting  these  values  above  and  reducing,  we  have 

r  P~\ 

/.  n  x  \r  +  iV  •  7 1 » (654^ 

and  making 


12r 


r   +    T7T~    "  T,  1 


we  obtain,  for  the  moment  of  the  friction  on  the  entire  ring, 

f-N.Ti (655) 

The  quantity  rt  is  called  the  mean  lever  of  friction  for  a  ring.  Since 
the    whole    friction    fN  may    be    considered    as    applied    at   a    point 

whose    distance  from    the    centre   is  §  R,  or  rt  =  r  -j-  tjt—  >    according 

l  «*  /* 

as  the  friction  is  exerted  over  an  entire  circle  or  over  a  ring, 
and  since  the  path  described  by  this  point  lies  always  in  the  di- 
rection in  which  the  friction  acts,  the  quantity  of  work  consumed 
by  it  will  be  equal  t3  the  product  of  its  intensity  fN  into  this 
path.  Designating  the  length  of  the  arc  described  at  the  unit's 
distance  from   C  by  st ,  the   path    in    q  lestion  will  be   either 

%Rst,     or     r,  v, 


444  ELEMENTS    OF    ANALYTICAI     MECHANICS. 

and    the    quantity  of  work  either 

%R.st.f.N 
for   an    entire   circle,    or 

for  a  ring.  Let  Q  denote  the  quantity  of  work  consumed  by  fric 
tion  in  the  unit  of  time,  and  n  the  number  of  revolutions  performed 
by  the  pivot   in   the   same    time ;    then  will 

s,  —  2  *  X  n ; 
and  we    shall    have 

Q  =  %«.R.f.  N.n (656) 

for   the   circle,  and 

Q  =  2*-f'N-  (r  +  — )  .  n      ....     (657) 

for    a   ring;   in  which  *  ==  3,1416. 

The  co-efficient  of  friction  /,  when  employed  in  either  of  the  fore- 
going  cases,  must   be    taken  from  Table  VI,  VII,  or  VIII. 

Example. — Required  the  moment  of  the  friction  on  a  pivot  of 
cast  iron,  working  into  a  socket  of  brass,  and  which  supports  a 
weight  of  1784  pounds,  the  diameter  of  the  circular  end  of  the 
pivot  being  6  inches.     Here 

in.  ft. 

R  =  «  =  3  =  0,25, 

lbs. 

N  =  1784, 
/  =  0,147 ; 
which,  substituted    in  Equation  (653),  gives 

lbs.  ft. 

0,147  x  1784  x  §  X  0,25  =  43,708. 

And  to  obtain  the  quantity  of  work  in  one  unit  of  time,  say  a 
minute,  there  being  20  revolutions  in  this  unit,  we  make  n  =  20, 
and  «r  =  3,1416  in  Equation  (656),  and  find 

Q  =  i  X  3,1416  X  0,25  x  0,147  x  1784  x  20  =  5402,80} 


APPLICATIONS.  445 

that  is  to  say,  during  each  unit  of  time,  there  is  a  quantity  of 
work  lost  which  would  be  sufficient  to  raise  a  weight  of  5492  80 
pounds  through   a  vertical   distance  of  one  foot. 

Example. — Required  the  moment  of  friction,  when  the  pivot'  sup- 
ports a  weight  of  2046  pounds,  and  works  upon  a  shoulder  whose 
exterior  and  interior  diameters  are  respectively  6  and  4  inches ;  the 
pivot   and   socket   being   of  cast   iron,  with  water   interposed. 

/  =  — - —  =  1  inch, 

r  =  2  +  0,5  .=  2,5  inches, 

(1)2  in.  ft. 

T*  =  2'5  +  12x2,5  =  2'5333  =  °'2111' 
N  =  2046  pounds, 
J  =  0,314; 
jrhich,  substituted  in  Expression  (655),  gives  for  the  moment  of  friction. 

0.314  x  2046  *'x  0,2111  =  135,62. 

The  quantity  of  work  consumed  in  one  minute,  there  being  sup- 
posed 10  revolutions  in  that  unit,  will  be  found  by  making  4n 
Equation  (657),  *  =  3,1416  and  n  =  10, 

Q  =  2  x  3,1416  x  0,314  x  2046  x  0,211  x  10  =  8517,24; 

that  is  to  say,  friction  will,  in  one  unit  of  time,  consume  a  quantity 
of  work  which  would  raise  8517,24  pounds  through  a  vertical  dis- 
tance of  one  foot.  The  quantity  of  work  consumed  in  any  given 
time  would  result  from  multiplying  the  work  above  found,  by  tho 
time   reduced   to   minutes. 

• 

TRUNNIONS. 

§361. — The  friction  on  trunnions  and  axles,  which  we  now  pio- 
ceed  to  consider,  gives  a  considerably  less  co-efficient  than  that  which 
accompanies  the  kinds  of  motion  referred  to  in  §  355.  This  will 
appear  from  Table  IX,  which  is   the   result  of  careful    experiment. 

The  contact  of  the   trunnion   with    its   box    is   along   a   linear  ele- 


m 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


merit,  common  to  the  surfaces  of  both.  A  section  perpendiculai  to 
its  length  would  cut  from  the  trunnion  and  its  box,  two  circles  tan 
gent  to  each  other  internally.  The  trunnion  being  acted  on  only  by 
it3  weight,  would,  when  at  rest,  give  this  tangential  point  at  o,  the 
lowest  point  of  the  section  p  o  q  of  the  box.  If  the  trunnion  be  put 
in  motion  by  the  application  of  a  force,  it  would  turn  around  the 
point  of  contact  and  roll 
indefinitely  along  the  sur- 
face of  the  box,  if  the 
latter  were  level ;  but  this 
not  being  the  case,  it  will 
ascend  along  the  inclined 
surface  op  to  some  point 
as  m,  where  the  inclina- 
tion of  the  tangent  urn  v 
is  such,  that  the  friction 
is  just  sufficient  to  pre- 
vent the  trunnion  from  sliding.  Here  let  the  trunnion  be  in  equili- 
brio.  But  the  equilibrium  requires  that  the  resultant  of  all  the 
forces  which  act,  friction  included,  shall  pass  through  the  point  m 
and  be  normal  to  the  surface  of  the  trunnion  at  that  point.  The 
friction 'is  applied  at  the  point  m;  hence  the  resultant  iV  of  all  the 
other  forces  must  pass  through  m  in  some  direction  as  m  d ;  the 
friction  acts  in  the  direction  of  the  tangent;  and  hence,  in  order 
that  the  resultant  of  the  friction  and  the  force  N  shall  be  normal  to 
the  surface,  the  tangential  component  of  the  latter  must,  when  the 
othe  r ,  component  is  normal,  be  equal  and  directly  opposed  to  the 
friction. 

Take  upon  the  direction  of  the  force  N  the  distance  m  d  to 
represent  its  intensity,  and  form  the  rectangle  ad  bm,  of  which 
the  side  m  b  shall  coincide  with  the  tangent,  then,  denoting  the 
angle  d<m  a  by  <p,  will  the  component  of  N  perpendicular  to  the  tan- 
gent'-be  '  :  v 
',....,/  iV.cosp; 

and  tjie  friction  due  to  this  pressure  will  be 

:  '     •  ■•"  /.  N '..  cos  (p. 


APPLICATIONS 


441 


The  component  of  N,  in  the  direction  of  the  tangent,  will  be 

W .  sin  9  ; 

and  as  this  must  be  equal  to  the  friction,  we  have 

/.  N .  cos  9  =  N .  sin  9  ; (658) 

whence, 

/  =  tan  9 ; 

that  is  to  say,  the   ratio   of  the  friction    to    the  pressure  on    the    trun- 
nion   is  equal    to    the    tangent  of  the   angle    which    the  direction    of  tht 
resultant  iV,  of  all  the  forces   except    the  friction,  makes   with    the    nor 
mal    to    the   surface    of    the    trunnion    at 
the  point  of  contact.     This  gives  an  easy 
method    of    finding     the    point    of   con- 
tact.     For   this   purpose,    we   have   but 
to   draw    through    the   centre   A   a   line 
A  Z,    parallel    to    the   direction    of    N, 
and    through    A    the    line   Am,  making 
with  A  Z   an   angle  of  which    the    tan- 
gent  is  /;    the  point   m,   in   which   this 
line    cuts     the    circular    section   of    the 

n 

trunnion,  will  be  the  point  of  contact. 

Because  madb,  last  figure,  is  a  rectangle,  we  have 

N2  =  N2  cos2  9  4-  N2  sin2  9  ; 

and,  substituting  for  N2  sin2  9  its  equal  f2  N2  cos2  9,  we  have 

N2  s=  iVr2cos29  +  /2iVr2cos29  =  JV2cos2q>  (1  +  Z3) ; 
whence, 


iVcos  9  =  y  x 


VTT?' 


and  multiplying  both  members  by  /, 

/ .  i\r .  cos  9  =  N  ' 


f 


(659) 


but   the    first   member    is   the     total    friction  ;    whence    we     conclude 
that   to  find  the  friction   upon  a   trunnion,  we  have  but  to  multiply  thr 


448 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


resultant  of  the  forces  which  act  upon  it  by  the  unit  of  friction,  found 
in  Table  IX,  and  divide  this  product  by  the  square  root  of  the  square 
of  this  same  unit  increased  by  unity. 

This  friction   acting  at  the  extremity  of  the  radius  R  of  the  trim 
nion  and  in  the  direction  of  the  tangent,   its  moment  will   be 

7 


if. 


-/i  +/2 


x  R. 


(660) 


Arid   the  path  described  by  the   point  of  application  of    the   friction 
being  denoted  by  Rs,,  the  quantity  of  work  of  the  friction  will  be 


N .  R  .  s,  x 


/ 


/M7/2' 


(661) 


in  which  s,  denotes  the  path  described  by  a  point  at  the  unit's  dis- 
tance from  the  centre  of  the  trunnion.  Denoting,  as  in  the  case  of 
the  pivot,  the  number  of  revolutions  performed  by  the  trunnion  in 
a  unit  of  time,  say  a  minute,  by  n ;  the  quantity  of  work  performed 
by  friction  in  this  time  by  Qt ;    and  making  t  ==  3,1416,  we  have 


and 


s,  =  2  at  .  n  ; 


Q   =  2«  .R.n.N. 


f 


VT+/2 


(662) 


When   the   trunnion   remains   fixed   and   does    not   form   part   of    the 
rotating   body,  the   latter  will    turn   about   the   trunnion,    which   now 
becomes    an    axle,   having    the   centre   of 
motion  at  A,  the   centre    of   the   eye   of 
the  wheel ;  in  this  case,  the  lever  of  fric- 
tion  becomes   the   radius    of    the    eye   of 
the    wheel.      As    the    quantity   of   work 
consumed    by    friction     is     the     greater, 
Equation     (662),    in    proportion    as    this 
radius   is   greater,   and   as   the   radius   of 
the    eye    of    the  wheel    must    be    greater 

ithan  that  of  the  axle,  the  trunnion  has  the   advantage,  in   this   respect 
over  the  axle. 


APPLICATIONS 


449 


The  value  of  the  quantity  of  work  consumed  by  friction  is  wholly 
independent  of  the  length  of  the  trunnion  or  axle,  and  no  advantage 
is  therefore  gained  by  making  it  shorter  or  longer. 


THE    CORD. 


§  362. — The  cord  and  its  properties  have  been  considered  in  paiL 
It  is  now  proposed  to  discuss  its  action  under  the  operation  of  fom-s 
applied  to  it  in  any  manner  whatever. 

Let  the  points  A\  A",  A"\  be  connected  with  each  other  by 
means  of  two  perfectly  flex- 
ible and  inextensible  cords 
A'  A",  A"  A'",  the  first 
point  being  acted  upon  by 
the  forces  P\  P",  &c. ;  the 
second  by  the  forces  Q\  Q'\ 
&c. ;  and  the  third  by  the 
forces  S\  £",  &c. ;  and  sup- 
pose  these  forces  to  be  in 
equilibrio.  Denote  the  co- 
ordinates of  A'  by  x'y'z', 
A"  by  *"  y"  z",  and  A'"  by 
*"'  y'"  z'".  Also,  the  alge- 
braic sum  of  the  components  of  the  forces  acting  at  A'  in  the  direo 
tion  of  xyz,  by  X'  T  Z ',  at  A"  by  X"  T'  Z",  and  at  Am  by 
X"  T"  Z"f.      Then  will,  §  101, 


X'    $x'    +  Y'    By'    +  Z'   Sz' 
+  X"  S  x"   +  Y"  S  y"  +  Z"  8  z"    }=  0. 
+  X"8x'"  +  Y'"8y'"  +  Z"'Sz'"  ) 

Denote  the  length  A'  A"  by  /,  and  A"  A"'  by  g;    then  will 


(663} 


L  =/-VP  -x'f  +  (y"  -y')2  +  (2"  -*')2  =  0; 


I-      .     (664) 


H  =  g-  <y/{?"  -  x"f  +  {y"'  -  y"f  +  (*"'  -  z"f  =  0. 

The    displacement    by  which  we   obtain    the    virtual    velocities    whose 


450  ELEMENTS    OF    ANALYTICAL    MECHANICS 

projections  are  8  x',  8y\  8 z\  dec,  is  not  wholly  arbitrary;  but  must 
be  made  so  as  to  satisfy  the  condition 

f/=0     and     5</  =  0. (665) 

Differentiating    Equations    (664),   and   writing    for   dx',    dy',    d  i' 
6  x\  8y\  8z\  &c,  we  find 

(*"  -  x')(8x"  -  8z')  +  (y"  -  y'){Sy"  -  fyQ  +  ^W)g  -  fe')  _  ft 

7  ~U; 

(x,,,^x,,){8xn,^8x,,)  +  (y',,-y,%8y,''-8yn)  +  {z,,,-z',){8zm-8z,f) 

These  being  multiplied  respectively  by  X'  and  X'",  and  added  to 
Equation  (663),  we  obtain  by  reduction,  and  by  the  principle  of 
indeterminate   co-efficients,  exactly  as  in  §213, 


X'  -  X'. 


x"  -  xf 


f 


=  0; 


X'.l JL  =0; 


/ 


Z'  —  X'  • 


z"  -  z' 


X"  +  X' . 


*"  -  *' 


/ 


/ 


-  X'"  • 


=  0; 


> 


(66C>) 


*"'  -  *" 


■> 


0; 


y»  +  x.JL—JL  _  fcw.L. — y_  =  0. 


/ 


Z"  +  X' 


•"  -  2' 


/ 


2'"   —    2" 

_  x'"  . —  =  0 


(667) 


X"'  +  X 


ttt 


x'"  -  x" 


=  0; 


y'"  —  y" 
Y»'  +  x'"  .  y- y—  -  0;    \ 


Z'"  +  X 


/// 


*'"  -  z" 


0; 


(668) 


Taking  from   each   group  its  first   equation  ar  d  adding,  and   doing 
the  same  for   the  second   and   third,  we   have 


X'  +  X"  +  X"l  =  0 ; 
Y'  +  Y"  +  Y'"  =  0 ; 
Z'  +  Z"  4-  Z'"  =  0. 


(669) 


AFPI  ICATIONS. 


451 


That   is,    the   conditions    of    equilibrium  of    the  forces   are,    §  80,    the 
same   as   though   they  had   been   applied  to   a   single   point. 

To  find  the  position  of  the  points,  eliminate  the  factors  X'  and 
X'",  and  for  this  purpose  add  the  first,  second  and  third  equations 
of  group  (667)  to  the  corresponding  equations  of  group  (668),  and 
there  will    result 

X"  +  X"'  +  y  (x"  -  x')  =  0 ; 

T'  +  Y"  +  —'V'  -  jfO  =  0; 

/ 

Z"  +  Z'"  +  h.  (Z"  -  Z')  =  0. 
irom  which  we  find  by  elimination, 


*LJ  L?L  (X"  +  X'")  =  0  ; 


*"  -  x- 


Z"  +  2 


nt 


z"  -  z' 


x"  -  X 


-  (X"  +  X"')  =  0. 


\ 


(670) 


From   group  (666),  by  eliminating  X', 


r-^, ^^'  =  0; 


X      —  X 


Z'  - 


z"  -  z' 


x"  -  X 


7^  =  0; 


(671) 


and    finally  from   group   (668)  we   obtain,  by    eliminating   X"', 


\rttr 

^"^ 

y"' 

— 

y" 

l 

Z'" 

x'" 

z'" 

x" 
z" 

X"' 

— 

x" 

.  X'"  = 


.  X'"  = 


0; 

0. 


•        •        •        • 


(672) 


Equations  (669),  (670),  (671)  and  672),  involve  all  the  conditions 
necessary  to  the  equilibrium,  and  the  last  three  groups,  in  connection 
with  group  (664),  determine  the  positions  of  the  points  A',  A" 
and  A'",  in  space. 


§  363. — The    reactions    in    the    system    which   impose  conditions  oil 


462         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

the   displacement   will    be    made    known   by   Equation    (331),   which 
because 

f       dL      V     r      dL       V     r       dL      -i* 

ld{x"-x')\  "1"Lrf(y"-y')J     U(*"-*')J  -1; 
U(x'" - x")\  T ly(y'" - y")\     \d (*'" - *")J        ' 

becomes  for  the   cord  A'  A"% 

V  =  JP ; 
and  for  the  cord  A"  A'", 

X'"  =  2V"  ; 

from  which  we   conclude,  that   X'  and   X'"   are   respectively  the   ten 
sions  of  the   cords  A'  A"  and  A"  A'". 

This   is   also    manifest  from    Equations  (666)    and   (668) ,    for,  by 
transposing,  squaring,  adding  and   reducing   by  the   relations, 


(*"  -  x'f  +  (y"  -  y')2  ±  {?"  -  z'f 

P 

* 
we  have 


=  1, 

g2 


X'    =  y/X'i     +   Y'2     +  Z'2     =  i?', 


(673) 


x/"  =  yx'"2  +  y"2  +  z"'2  =  .#'", 

in  which  i?'    and  R'"  are   the    resultants  of    the    forces   acting   upon 
the  points  A'  and  A'"  respectively. 

Substituting   these  values   in   Equations  (666)  and  (668),  we  have 

£!  -  *"  ~  *'     IL  -  y"  ~" y'    E.  -  z"  r  *' . 
^  ~     7     ;  IB  ~     7    ;  ]sP  ~     7~ ' 

X'"  x'n  —  x"      T"  _        y'"  —  y"      Z"'  z'"  —  z" 

ill777  ~  a        ;    ~W*  =  a        '    R 


ttt 


whence  the  resultants  of  the  forces  applied  at  the  points  A'  and  A"'t 
act  in  the  directions  of  the  cords  connecting  these  points  with  the 
point  A",  and  will  be  equal  to,  indeed  determine  the  tensions  of 
these    cords. 


APPLICATIONS 


453 


§  364. — From    Equations    (669),    we    have    by    transposition, 
X"  =  -  (X'"  +  X')  ;    Y"  =  -  ( Y"'  +  Y') ;    Z"  =  -  (Z"f  4-  £')• 

Squaring,    adding   and    denoting   the    resultant   of    the   forces   applied 
at  A"  by  R",  we  have 


R"  =  ^(X"'  +  X')2  +  ( Y"'  +  F')2  +  (Z"f  -h  Z')2  '  *  (6?4) 
and    dividing  each  of  the   above   equations   by  this  one 


X" 
R" 

Y" 
R" 

Z" 
~RT 


X"'  +  X 


t     ^ 


R" 

> 

Y"  + 

r7 

• 

R" 

? 

Z"'  4- 

Z' 

• 

i? 


it  y 


(675) 


whence,  Equation  (674),  the  resultant  of  the  forces  applied  at  A"  is 
equal  and  immediately  opposed  to  the  resultant  of  all  the  forces 
applied   both  at  A'  and   A"' 

If,  therefore,  from  the  point 
A",  distances  A"  m  and  A"  n 
be  taken  proportional  to  R'  and 
R'"  respectively,  and  a  paral- 
lelogram A"  m  Cn  be  constructed, 
A"  C  will  represent  the  value  of 
R".  If  A' A"  A"'  be  a  contin- 
uous cord,  and  the  point  A" 
capable  of  sliding  thereon,  the 
tension  of  the  cord  would  be 
the  same  throughout,  in  which 
case  R'  would  be  equal  to  R'", 
and  the  direction  of  R"  would 
bisect  the  angle  A'  A"  A"'. 

The    same    result  is  shown  if, 

instead    of    making    Sf  =  0   and 

8  g  =  0     separately,     we     make 

28 


454  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

^  (/  1"  9)  —  0>    multiply    by    a     single     indeterminate     quantity    X, 
and  proceed    as    before. 

§  365. — Had  there  been  four 
points,  A',  A",  A'"  and  A", 
connected  by  the  same  means, 
the  general  equation  of  equili- 
brium would  become,  by  call- 
ing h  the  distance  between  the 
points,  A'"  and  Aiy, 

X'  8  x'  +  X"  8  x"  +  X"  8  x'"  4-  ^iv  *  ***  1 
•  4-  Y'  I  y'  4-  Y"  8  y"  4-  Y"  8  y"'  +  Fiv  8  yiv 
+  Z'  6  z'  4-  Z"  8  z"  4-  £'"  *  z'"  +  Z"  S  z" 
4-  X'  8/  4-  X"  8y     4-   X'"<U 

and    from  which,  by  substituting    the  values  of  8  f%  8g    and  8  h,  the 
folic  wing  equations  will  result,  viz.  : 


0; 


X  -  X'  . 


*"  —  xf 


T    =°' 


x' . y"  =  y/  =  0, 


/ 


v. 


Z"    -    2' 


f 


-0, 


(676) 


X'  4-  X' 


x»  -  x>                or'"  _  x» 
X     • 


Y"  +   X'  •  2-    -r-^-    -    X 


Z"  4-  X' . 


X"'  4-  X 


// 


Z"'  4-  X" 


/ 

f  - 

y' 

f 

z"   - 

<_ 

/ 

x'"  - 

x" 

9 

y"'  - 

y" 

9 

z,„  _ 

z" 

y>»    _   y» 


Jit 


Jf 


—  X 


// 


=  0, 

=  0, 
=  0, 


—  X 


Itt 


a:IT  —  x 


tit 


=  0, 


,nt 


Y'"  4.  X"  •  Z 1 X'"  .  ylT   t   y      =  0, 


h 


X 


f/> 


2iv     —  2'" 


A 


=  0, 


(677) 


>     '     (678) 


APPLICATIONS 


IK 


xy*  —  x 


tt> 


X" 

+ 

V" 

—  0 

h     -0' 

y\r 

+ 

X'" 

yiv   _   yf" 

Z,r 

+ 

X'" 

■                       —  0 

I.         —  u> 

(679) 


Eliminating  the  indeterminate  quantities  X^  X"  and  X'",  we  obtain 
eight  equations,  from  which,  and  the  three  equations  of  conditions 
expressive  of  the  lengths  of  /,  g,  and  A,  the  position  of  the  points  A', 
A",  A"',  and  AiY  may  be  determined. 

If  there  be  n  points,  connected  in  the  same  way  and  acted  upon 
by  any  forces,  the  law  which  is  manifest  in  the  formation  of  Equa- 
tions (676),  (677),  (678),  and  (679),  plainly  indicates  the  following 
r  equations  of  equilibrium  : 


Y7 

x"  —  x' 

>'  ■                    —  0 

A.\ 

A.    •           j.           —  v/, 

T 

-  X'  . fLjzX  =  o, 

Z' 

z"  —  z' 
V  •  -      .         -  0, 

/' 


(680* 


X"  +  X' 


*"  -  *' 


/ 


—  X 


ft 


x'"  -  x" 


=  0, 


r  +  vJ-LzlL-vJ"-*9^ 


f 


Z"  +  x' 


z"  -  *' 


n 


9 

— 


/ 


=  0, 


(f«M) 


X'"  +  X 


tit  ~n 

-  X 


•> III    x    —  x 


.III 


.11 


y '      y       \ttt  y       y 


Y"'  -f  X' 


—  X1 


Z'"  +  x 


mill       Z" 

tt  .  * f_    y^ilt 


h 

z"  —  z' 


h 


0, 
0, 
0, 


=  0.   V 


(682) 


450  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


Xm_,  +  X._8  .  «-'  7  *—  -  X„_,     «■  '     *~* 


r.i,  +  \._8 


A 

y. 

-i 

— 

y- 

-8 

k 

*— 

-i 

— 

«»- 

-1 

-  V-. 


i 

y. 

—  y- 

-l 

/ 

2. 

—  *.- 

•  1 

=  o, 


=  0, 


Z.-i  +  *.-.  •    ""•    ,    ""'  ■      X..,  .  2--^=i  =  o, 


x.  +  x.,,.*--,**-'^,] 


(683) 


r.  +  x*-, 


£„   +X... 


y»  —  r— 


!  =  6, 


>     »    • 


z._ 


m-1 


/ 


=  0. 


(684) 


In  which  X,  with  its  particular  accent,  denotes  the  tension  of  the 
cord  into  the  difference  of  whose  extreme  co-ordinates  it  is  multi- 
plied. 

Adding  together   the   equations   containing   the   components  of  the 
forces  parallel  to  the  same  axis,  there  will  result 


X9  +  Jg"  +  X"  +  I1T    .    .    .    Xn  =  0,  } 

Y>  +  F"   f-  Y'"  +  Tir    •    •    •     Ym  =  0,  I  •     • 

Z'  +  Z"  +  3'"  +  £iT     •    •    -    z%  =  0,  J 


(685) 


from    which   we    infer,    that     the    conditions   of    equilibrium    are   the 
same  as  though  the  forces  were  all   applied  to  a  single  point. 

From   group  (680),  we   find   by  transposing,  squaring,  adding  and 
extracting   square   root, 


<y/X'2  +  r2  +  Z'2  =  X'  =  R 

and  dividing  each  of  the  equations  found  after  transposing   in   group 
(680)  by  this   on  3, 


Bf 

H 
R' 


jt 


-x> 


f 


y"  -  y' 


£  _  *"  -  z' 

B!~        f 


APPLICATIONS. 


457 


Treating    the    equations  of    group    (684)    in    the     3ame    way,    we 
have 


R. 


R. 


R. 


*» 

—  #•- 

-1 

/ 

Vn 

—  Uk- 

-l 

l 

*» 

—  *m- 

-I 

/ 


whence,  the  resultants 

of  the   forces   applied 

to  the  extreme  points 

A'  and  An,  act  in  the 

direction  of  the  extreme  cords.     And  from  Equations  (685)  it  appears 

that    the   resultant   of  these   two  resultants   is  equal   and   contrary  to 

that    of  all   the   forces   applied    to    the    other   points. 


§366.— If  the  extreme  points  be  fixed,  X\  Y\  Z'  and  Xn,  Yn,  Zn, 

will  be  the  components  of  the  resistances  of  these  points  in  the 
directions  of  the  axes ;  these  resistances  will  be  equal  to  the  ten- 
sions \'  and  Xn  of  the  cords  which  terminate  in  them.  Taking  the 
sum  of  the  equations  in  groups  (680)  to  (684),  stopping  at  the  point 
whose    co-ordinates   are  #,»_»,  y^_m,  s»_m>  we   have 


X>  +  U-^ 


r  +  u-u.,. 


Z'  +  SZ-X^,.,. 


Xn— m  Xn—r*—\ 


Vn—m    Vn-m-\ 


Z  Z 


0; 
0; 
0; 


(686) 


in  which  2  Xy  2F,  2  Z,  denote  the  algebraic  sums  of  the  components 
in  the  directions  of  the  axes  of  the  active  forces;  Xw_lir_1  the  tension 
on  the  side  of  which  the  extreme  co-ordinates  are  ar,^.,  #»_,»,  *»_», 
and  s„_*_i,  y»_ ^_i,  K-m-\\   an(i  *»—  tne  length  of  this   side. 


§367. — Now,    suppose   the    length   of    the    sides    diminished    and 


458 


ELEMENTS     OF     ANALYTICAL     MECHANICS. 


their    number    increased    indefinitely ;    the    polygon    will     become    a 
curve;    also,  making   '^t^m-l  =  t1  we   have 

y-m  —  y»-«-i  ==  dy, 

K~m  —  *»-m-l   =  dz, 

Cm    =    dS, 

i  being   any   length  of  the   curve ;  and  Equations  (686)  become 

dx 

I'  +  U-/.-=0; 

d  s 


as 

a*2 
a  s 


(687) 


which  will  give  the  curved  locus  of  a  rope  or  chain,  fastened  at 
its  ends,  and  acted  upon  by  any  forces  whatever,  as  its  own  weight, 
the  weight  of  other  materials,  the  pressure  of  winds,  curreuts  of 
water,  &c,  &c. 

This  arrangement  of  several  points,  connected  by  means  of  flexi- 
ble cords,  and  subjected  to  the  action  of  forces,  is  called  a  Funi- 
cular Machine. 

• 

§368. — If  the  only  forces  acting  be  pressure  from  weights,  we 
have,  by  taking   the   axis   of  z  vertical, 

X"  =  X"  =  X  T  &c,  =  0  ;     Y"  =  Y'"  &c  =  0 ; 
and    from    Equations  (680)   to   (684), 


X    sz  X' 


x"  -  x1 


f 


=  X 


n 


x'"  -  *" 


•      •      •      • 


Xn  —  £„_, 

x-1      TT  "*' 


whence,  the  tensions  on  all  the  cords,  estimated  in  a  horizontal 
direction,  are  equal  to  one  another.  Moreover,  we  obtain  from  the 
*ame    equations,    by  division, 


y"  -  y' 

x"  -  x- 


y'"  -  y" 

xm  -  x" 


y.  -  y»-i 

—  • 


APPLICATIONS. 


450 


These  are  the  tangents  of  the  angles  which  the  projections  of  the 
sides  on  the  plane  xy  make  with  the  axis  x.  The  polygon  is 
therefore   contained  in   a   vertical   plane. 


THE   CATENARY. 


§369. — If  a  single  rope  or  chain  cable  be  taken,  and  subjected 
only  to  the  action  of  its  own  weight,  it  will  assume  a  curvilinear 
shape  called  the  Catenary  curve.  It  will  lie  in  a  vertical  plane. 
Take  the  axes  z  and  x  in  this  plane,  and  z  positive  upwards,  then 
will 

U=0;     2F=0;      F'  =  0;     2Z=-W; 

in    which    W  denotes   the  weight  of  the   cable,  and    Equations  (687) 
become 


dx 
X'-l--  =0, 
as 

d  z 

zf  -w  -t.—  =0. 

d  s 


(688) 


Z 


These  are  the  differential  equations  of  the  curve.  The  origin 
may  be  taken  at  any  point. 
Let  it  be  at  the  bottom  point 
of  the  curve.  The  curve 
being  at  rest,  will  not  be 
disturbed  by  taking  any  one 
of  its  points  fixed  at  pleas- 
ure. Suppose  the  lowest 
point  for  a  moment  to  be- 
come   fixed.      As    the    curve 

is   here  horizontal,  Z'  =  0,  §  366,  and  from    the   second  of  Equations 
(688),  we   have 

dz 


W  = 


ds' 


(689) 


whence,  the  vertical  component  of  the  tension  at  any  point  as  0  of 
the  curve,  is  equal  to  the  weight  of  that  part  of  the  cable  between 
this  point  and  the  lowest  point.     The  first  of  Equations  (688)  shows 


460  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

that  the  horizontal  component  <»f  the  tension  at  0  is  equal  to  the 
tension  at  the  lowest  point,  as  it  should  be,  since  the  horizontal 
tensions   are  equal  throughout. 

Taking   the   unit   of  length  of  the  cable   to  give  a   unit  of  weight, 

* 

which  would  give  the  common  catenary,  we  have  W  =  s  ;  and,  de- 
noting' the    tension  at    the  lowest   point  by  c,  we  have 


t  =   =fc  -y/s2  +  c2, 
and  from  Equation  (689), 

s  •  ds 

dz  =  q=  —  • 


y/c2  +  •* 

Taking  the  positive  sign,  because  z  and  s  increase  together,  inte- 
grating, and  finding  the  constant  of  integration  such  that  when 
z  —  0,  we   have   s  =  0, 

z  -\-  c  =.  y/  c2  -j-  6'2  ; 

whence, 

,?2  a  z2  -f  2  c  2. 

Also,  dividing  the  first   of  Equations  (688)  by   Equation  (689), 

dx  c  c 

dz  s  y^2  +  2cz  ' 

and  integrating,  and  taking  the  constant  such  that  x  and  z  vanish 
together, 

*  =  ,.log  i±c+y*  +  2c±   ;    ;    ;   (690) 

■ 

which    is   the   equation  of  the  catenary. 

This   equation    may    be   put    under    another   form.      For  we    mav 

write    the   above, 

• 

c  e^  =  z  +  c  +  y/{?  +  c)  2  —  c2 ; 

transposing  z  4-  c  and   squaring, 

c2  >  ee    —  Z  c  ee  (z  -f  c)  =  —  c2  ; 
whence, 

?  +  «  =  |c    >£  -j-  <?~7). (691) 


APPLICATIONS. 


161 


Also, 


and   by    substitution, 


=  V(«  +  <)*  - 


*  =  $c-(ee  —  e    e).     ...........     (692)1 

§370. — If  the  length  of  the  portion  of  the  cable  which  gives  a 
unit  of  weight  were  to  vary,  the  variation  might  be  made  such  as 
N>  cause  the  area  of  the  cross  section  to  be  proportional  to  the 
tension  at  the  point  where  the  section  is  made.  The  general  Equa- 
tions (638)  will    give    the    solution   for   every  possible  ease. 

FRICTION    BETWEEN    CORDS    AND    CYLINDRICAL    SOLIDS. 


§  371. — When  a  cord  is  wrapped  around  a  solid  cylinder,  and 
motion  is  communicated  by  applying  the  power  F  at  one  end 
while  a  resistance  W  acts  at  the  other,  a  pressure  is  exerted  by 
the  cord  upon  the  cylinder ;  this  pressure  produces  friction,  and  this 
acts  as  a  resistance.  To  estimate  its  amount,  denote  the  radius 
of  the  cylinder  by  is?,  the  arc  of  contact  by  »,  the  tension  of  ths 
cord    at   any  point  by  t. 

The  tension  t  being  the  same 
throughout  the  length  d  s  —  a  tt 
of  the  cord,  this  element  will  be 
pressed  against  the  cylinder  by 
two  forces  each  equal  to  /,  and 
applied  at  its  extremities  a  and  tt , 
the  first  acting  from  a  towards 
W,  the  second  from  tt  towards  b'. 
Denoting  by  &  the  angle  a  b  tt, 
and  by  p  the  resultant  brn  of 
these  forces,  which  is  obviously 
the    pressure  of  ds  against   the   cylinder,  we   have,  Equation  (56), 


p  =  y/P  +  t*  +  2  t . :  cos  6  =  t  y^2(l  +  cos  6) ; 


but 


1  -J-  cos  6  ==  2  cos2  £  6  ; 


(t  -  »)  =  -g  ; 


462  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and    taking   the   arc   fur    its    sine,    because   n  —  6   is   very   small,    we 

have 

ds 

and   hence,  8  355.  the  friction  on  ds  will  be 

>  •    ,    ds 

f'P  =f't'-ft' 

The  element  tt  £2  of  the  cord  which  next  succeeds  a  tt  7  will  have 
its  tension  increased  by  this  friction  before  the  latter  can  be  over 
come  ;  this  friction  is  therefore  the  differential  of  the  tension,  being 
the    difference    of  the    tensions  of  two  consecutive   elements  ;  whence, 

dt=f.t---i 
dividing   by    t   and    integrating, 


or, 


log  t  =  f.±  +  log  C, 


t  =   CeR (693) 

making  s  =  0,  we   have  t  =  W  =   C;  whence, 

t  =  W*tT\ (694) 

and   making  «  =  &  =  al,  ^  f8,  we   have  t  =  F;  and 

F=  W-eR        (695) 

Suppose,  for  example,  the    cord    to    be  wound  around  the  .ylinder 
three   times,  and  /  =  ^  ;  then  will 

S  =  3«.2B  =-.  6  .  3,1416.  E  s  18,849 B, 
and 

^=Trx^X,884S=^X(2?71825)63838; 

or, 

I  =  W.  535,3, 

that  is  to  say,  one  man  at  the  end  W  could  resist  the  combined 
effort  of  535  men,  of  the  same  strength  as  himself,  to  put  the  cord 
in   motion  when   wound   three   times   round   the    cylindei. 


A  PPLtCATIONS 


463 


THE    INCLINED    PLANE. 

$  372. — The  inclined  plane  is  used  to  support,  in  part,  the  weight 
of  a    body  while    at    rest  or    in    motion    upon    its    surface. 

Suppose  a  hody  to  rest  with  one  of  its  faces  on  an  inclined  plane 
ni    which    the    Equation    is 

L  =  cos  a  x  -f  cos  b  y  -+-  cos  c  z  —  d  =  0  ;     •     •     •     •    (a) 

in  which  d  denotes  the  distance  of  the  plane  from  the  origin  of  co- 
ordinates, and  a,  b,  c,  the  angles  which  a  normal  to  the  plane  makes 
with    the    axes    x,  y,   z,  respectively. 

.Denote  the  weight  of  the  body  by  Wt  the  power  by  F  ;  the  nor- 
mal pressure  by  N ;  the  angles  which  the  power  makes  with  the 
axes  ./•,  //,  i,  by  a,,  /3,,  yt,  respectively;  and  the  path  described  by 
the  point  of  application  of"  the  resultant  friction  by  s.  Then,  taking 
ihe  axis  Z  vertical  and  positive  upwards,  and  supposing  the  force  to 
produce    a  uniform    motion  of  simple   translation,   will,   Eq.  (645), 

dx^ 


(Vcosa,  +/JV  -^)  Sx 


+  (F<KM&4+fNp?) 


d  s 
dif 


d  a 
rdz 


y 


\  -0; 


+  {Fcos  y4  +/N  ~  -  W 


)  Sz 


d  s 

and.    Equation    (a), 

cos  a  d  x  -f  cos  b  d  y  -f-  cos  c  o  2  =  0 
Multiplying     this    last    by    X,    adding    and    proceeding    as    in    §  213, 

dx 


F  cos  a    -\-  f  N h  X  cos  a  =  0, 

d  s 

F  cos  ft,  +  fff  %2 -fc  X.oob*  =  0, 

a  s 

F  cos  y,  +  f  N  -^  +  X  cos  c  —  W  ~  0  ; 


W 


and,  Eq.  (331), 

XT  ■>  / 

f^yfe)  +W  r(rfl)=x- 


r/  /A" 


■dL\* 


(d) 


Substituting  the  vai  le  of  X  in    Equations  ('„),  the   tirst    two  give  by 
eliminating    iV", 


464        ELEMENTS    OF    ANALYTICAL    MECHANICS. 


d  x 

f (-  cos  a 

ds  cos  p, 

-  —       — ...    -  *       —  — 

dy  ,       cos  a 

/  — -  -h  cos  b 
ds 


+  1  = 


{*) 


I 


and    th3   first    and    third,  bj   eliminating  N. 

I  I//  cos  v,—  —  cos  a,—  )  +  cos  7.  cos  a  —  cos  a  cos  c  \—W(  f  —  -f  cos  «  )  . 
i_  \      "ds  'dsl  :       J  V  efc  //a 

If  there    be    no    friction,  then    will  fss  0,  and,  Eq.    (e), 

cos  a    cos  # 

i  +  1=0; 

cos  0    cos  at 

whence,  Eqs.    (45)    and   (a),   the    power    must    he    applied    in    a    jiaiir 
iioimnl    hoth   to    the    inclined    plane    and    to   the    horizon. 

If  without  disregarding  friction,  the  power  be  applied  in  a  plane 
fulfilling-  the  above  condition,  and  also  con- 
taining  the  centre  of  gravity,  the  resultant 
friction  may  be  regarded  as  acting  in  this 
plane,  and  we  may  take  it  as  the  co- 
ordinate   plane   z  x,    in    which    case 

cos  b  =  0 ;  cos  (3   =  0  ;  --  —  0 ; 

d  t 

and    denoting    the    inclination    of  the    plane    to   the    horizon  by    a,    and 
that   01    thi    power    to   the    inclined    plane    by  <p  ; 


dx 


cos  a  =  sin  a  ;  cos  c  :=  —  cos  a  ;  cos  ryi  =  sm  ay ; 
dz 


cos  yt  — cos  v.t  — —   =  —  sin  a,  cos  a  -f-  cosa^  sin  a  =  sin  (a  —  ay)  =  sin  <p  ; 


d  s 


d  s 


cos  y t  cos  #  —  cos  a.t  cose  =  sin  ay  sin  a  -f-  cos  ay  cos  a  =  cos  (a  —  a,)  =  cos<p ,' 
which,  in   Eq.  (g),  give 


„        W  (sin  a  -f  /  co  3  a) 
cos  <p  -f-  /sin  <p 


(696) 


This  supposes  motion  to  take  place  up  the  plane  ;  if  the  power  F 
be  just  sufficient  to  permit  the  body  to  move  uniformly  down  the 
plane,  then  will  /  change  its  sign,  and  we  shall  have 

cos  9  — /sin  <p 

And  the  power  may  vary  between  the  limits  given  by  these  two 
values  without   moving  the  body. 


i  PPLICATIONS.  465 

§  3*73.  If  the  power  be  zero,  or  F  =  0,  then  will 

sin  a  —  /  cos  a  =  0, 

tan  a  =  /, 
which  is  the  angle  of  friction,  §  355. 

§  374. — If  the    power   act   parallel    to   the  plane,  then  will  q>  =s  0, 

and 

F—  JF(sina  ±/cos  a) *  .     (698) 

the  upper  sign  answering  to  the   case  of  motion    up,  and    the    lower, 
down  the  plane;    the  difference  of  the  two  values  being 

2/  JFcosa. 
If  /=  0,   then    will 

F         .  £  C 

that  is,  the  power  is  to   the  weight  as  the  height  of  the  plane  is  to 
its  length;    and   there   will   be  a  gain  of  power. 

§  375. — If  the  power  be  applied  horizontally,  then  will  <p  be  nega- 
tive and  equal  to  a,  and  we  have,  by  including  the  motion  in  both 
directions, 

r=  IT  (»in  «*/«*«) 

cos  a  qp  f  sin  a  v        ' 

tne  difference  of  the  limiting  values  being 

2/.  W 

cos2  a  —  f2  sin2  a* 

If  the  friction  be  zero,  or  /  =  0,  then  will 

F  BC 

That    is,  the    power  will    be    to    the    resistance    as    the  height  of  the 
plane  is  Jo  its  base;    and    there   may    be  gain    or   loss   of  power. 

§  376. — To  find  under  what  angle  the  power  will  act  to  greatest 
advantage,  make  the  denominator  in  Equation  (696)  a  maximum. 
For  this  purpose,  we  have,  by  differentiating, 

—  sin  <p  +  /cos  <p  =  0  j 


m 


ELEMENTS    OF    ANALYTICAL    MECHANICS 


L. 

h*/ 

b! 

B 

1 

rf 

(* 

h 

c* 

A     t 

\ 

M 

l 

V 

whence, 

tan  <p  =  f. 

That  is,  the  angle  should   be   positive,  and  equal  to  that  of  the  fric- 
tion. 

g  377. — If  the  power  act  parallel  to  any  inclined  surface  to  move 
a  body  up,  the  elementary  quantity  of  work  of  the  power  and  resists 
ances  will  give  the  relation,  Equation  (698), 

F d  s  =  Wds  sin  a  +  Wf  d  s  cos  a. 

But,    denoting    the    whole    hori- 
zontal   distance   passed   over   by 


/  =  A  (7,  and  the  vertical  height 
by  h  =  B  (7,  we  have 

d  s  .  sin  a  =  d  A, 

d  8  .  cos  a  =  d  I ; 

whence,  substituting,  and  integrating,  and  supposing  the  body  to  be 
started  from  reet  and  brought  to  rest  again,  in  which  case  the  work 
of  inertia    wiL    ualance  itself,  we  have 

Fs  =  Wh  4-  /.  W.  /, (700) 

in  which  there  is  no  trace  of  the  path  actually  passed  over  by  the 
body.  The  work  is  that  required  to  raise  the  body  through  a  ver- 
tical height  B  C,  and  to  overcome  the  friction  due  to  its  weight  over 
a   horizontal    distance    A  C. 

The  resultant  of  the  weight  and  the  power  must  intersect  tne 
inclined  plane  within  the  polygon,  formed  by  joining  the  points  jf 
contact  of  the  body,  else  the  body  will  roll,  and   not  slide. 

THE    LEVER. 

§  378. — The  Lever  is  a  solid 
bar  A  B,  of  any  form,  supported 
by  a  fixed  point  0,  about  which 
it  may  freely  turn,  called  the  ful- 
crum. Sometimes  it  is  supported 
upon     trunnions,    and    frequently 


APPLICATIONS. 


467 


upon  a  knife-edge.  Levers  have 
been  divided  into  three  different 
classes,  called  orders. 

In  levers  of  the  first  order,  the 
power  F  and  resistance  Q  are 
applied  on  opposite  sides  of  the 
fulcrum  0\  in  levers  of  the  second 
order,  the  resistance  Q  is  applied 
to  some  point  between  the  ful- 
crum 0  and  the  point  of  appli- 
cation of  the  power  F\  and  in 
the  third  order  of  levers,  the 
power  F  is  applied  between  the 
fulcrum  0  and  point  of  applica- 
tion of  the  resistance   Q. 

The  common  shears  furnishes 
an  example  of  a  pair  of  levers 
of  the  first  order ;  the  nut-crackers 
of  the  second ;  and  fire-tongs  of 
the  third.  In  all  orders,  the  con- 
ditions of  equilibrium  are  the 
same. 

These  divisions  are  wholly  ar- 
bitrary, being  founded  in  no  dif- 
ference of  principle.  The  relation 
of  the  power  to  the  resistances, 
is   the  same   in  all. 

Let  A  B  be  a  lever  supported 
upon  a  trunnion  at  0,  and  acted 
upon  by  the  power  P  and  resist- 
ance Q,  applied  in  a  plane  per- 
pendicular to  the  axis  of  the  trun- 
nion. Draw  from  the  axis  of  the 
trunnion,  the  lever  arms  On  and 
Om,  being  the  perpendicular  dis- 
tances of  the  power  and  resistance 
from     the    axis    of     motion,    and 


0 


o^O 


'Q 


468  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

denote    them   respectively  by  lp  and  lr;  also   denote  the   resultant  of 
P  and   Q  by  iV,  the   radius  of  the   trunnion   by  r,  the  co-efficient  of 
friction   by  /,  and   the   arc   described   at  the  unit's   distance  from    the 
■  m  is   by  5, . 
Then, 

$p  s=  ^ .  d  *} ;     d  q  =  lr .  d  Si , 

N=  tfF*  +  Q*  +  2?  Qoo&i, 

in  which  d  is  the  angle  of  inclination  A  C  B  of  the  power  to  the 
resistance.  Then,  supposing  the  lever  to  have  attained  a  uniform 
motion,  will,  Equations  (645)  and  (661), 

P.L.dsx-Q.l,.ds,-JP2  +  Q*  +  2PQcos6>    '-^==£2  as  O.ttOl] 

Omitting  the  common  factor  rf«, ,  and  making 

=  /;    m  =  r;    w  = 


we  have, 


P  -  m  Q  -  ^P2  +  §2+2/)  £  .  cos  0  •  /'»  =  0. 
Transposing,  squaring,  and  solving,  with  respect  to  P,  we  find, 


*=  « = " l-f>2T? " Li70V 

If  the  fraction  n  be  so  small  as  to  justify  the  omission  of  every 
term  into  which  it  enters  as  a„factor,  or  if  the  co-efficient  of  friction 
be  sensibly  zero,  then  would 

i=m=i (703) 

That  is,  the  power  and  the  resistance  are  to  each  other  inversely  as 
the  lengths  of  their  respective  lever  arms. 

If  the  power  or  the  resistance,  or  both,  be  applied  in  a  plane 
oblique  to  the  axis  of  the  trunnion,  each  oblique  action  must  be 
replaced  by  its  components,  one  of  which  is  perpendicular,  and  the. 
other  parallel  to  the  axis  of  the  trunnion.  The  perpendicular  com- 
ponents must  be  treated   as  above.     The  parallel  components  will,  if 


APPLICATIONS. 


469 


the  friction  arising  from  the  resultant  of  the  norma]  components  be 
not  too  great,  give  motion  to  the  whole  body  of  the  lever  along  the 
trunnion  :  and  if  this  be  prevented  by  a  shoulder,  the  friction  upon 
this  shoulder  becomes  an  additional  resistance,  whose  elementary 
quantity  of  work  may  be  computed  by  means  of  Eq.  (657)  and  made 
another  term  in  Equation  (701). 


WHEEL   AND   AXLE. 

§379. — This  machine  consists  of  a  wheel ,  mounted  upon  an  arbor, 
supported  at  either  end  by  a  trun- 
nion resting  in  a  box  or  trunnion 
bed.  The  plane  of  the  wheel  is  at 
right  angles  to  the  arbor ;  the  pow- 
er P  is  applied  to  a  rope  wound 
round  the  wheel,  the  resistance  to 
another  rope  wound  in  the  opposite 
direction  about  the  arbor,  and  both 
act  in  planes  at  right  angles  to  the 
axis  of  motion.  Let  us  suppose  the 
arbor  to  be  horizontal  and  the  re- 
sistance  Q  to  be  a  weight. 

Make 
N  and  N'  =  pressures  upon  the  trunnion  boxes  at  A  and  B ; 
R  =  radius  of  the  wheel ; 
r  =  radius  of  the  arbor ; 
p  and  p'  =  radii  of  the  trunnions  at  A  and  B ; 

f 


/'  = 


-/i  +p 


s,  =  arc   described    at    unit's  distance    from  axis  of   motion 
Ihen,  the  system  being  retained  by  a  fixed  axis,  we  have 

P  Sp  =  PRd  sx\ 
Q  8  q  =  Q  r  d  8V 

The  elementary  work  of  the  friction  will,  Eq.  (661),  be 


f(2tf+  N'P')dsri 
29 


470  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and     the    elementary    work    of    the     stiffness    of    cordage.    Equation 
(652), 

,     K+I.  Q 
d4  •  — r.dSi; 

and  when  the  machine  is  moving  uniformly, 

PRdsx-Qrdsx-f{N?  +  N'?,)dsl-dr:^r^-^..r'dsl  =  0',  .  (704) 

&  v 

The  pressures  N  and  N'  arise  from  the  action  of  the  power  P.  the 
weight  of  the  machine,  and  the  reaction  of  the  resistance  Q,  in 
creased  bv  the  stiffness  of  cordage.  To  find  their  values,  resolve 
each  of  these  forces  into  two  parallel  components  acting  in  planes 
which  are  perpendicular  to  the  axis  of  the  arbor  at  the  trunnion 
beds*  then  resolve  each  of  these  components  which  are  oblique  to 
the  components  of  Q  into  two  others,  one  parallel  and  the  other 
perpendicular  to  the  direction  of  Q. 

Make 
w   =  weight   of  the   wheel   and   axle, 
g    =  the   distance   of  its   centre    of  gravity   from  A, 
p    =  the   distance  m  A, 
q    —  the   distance   n  A, 
I    ■=:  length   of  the   arbor    A  B. 

9    =s  the   angle    which    the    direction    of  P   makes    with    the    vertical 
or   direction    of    the   resistance    Q. 

Then    the    force   applied    in    the   plane  perpendicular    to    the    trunnion 
A.  and  acting  parallel  to  the  resistance   Q,  will,  §  95,  be, 

and  the  force  applied  in  this  plane   and  acting  at  right  angles  to  the 
direction  of  Q,  will  be 

P j-^-  •  sin  <p. 

c 

The  vertical  force  applied  ir.  the  plane  at  B  will  be 

W  '  f  +  9  '  T  +  P  '  f  "  cos  *» 


APPLICATIONS.  471 


•»nd  the  horizontal   force  in  this  plane  will  be 


P  •  j  '  sin  9  ; 


whence, 


y  =  jV  [w(/-^)+C(/-j)  +  />(^-/»)cos(pia  +  iw(/-i>)».8iii»(p;     (705) 

JST=lrJ[w.g  +   £.?  4-  P.jo.coscp]2  +  P*  .  p*  .  sin*  9  ;  .   -(706) 

If  d  and  &'  be  the  angles  which    the  directions  of  JV  and  ATf  make 
with  that  of  the  resistance   Q,  we  have 

.  P(l-P)      .  .    At        Pp      . 

sin  0  =  — — .— -  •  sin  9  ;     sin  r  =  — —  •  sin  9. 

iV\  /  N'l  r 

Equations  (704),  (705),  and  (706)  are  sufficient  to  determine  the  rela- 
tion between  P  and  Q  to  preserve  the  motion  uniform,  or  an  equili- 
brium without  the  aid  of  inertia.  The  values  of  iV  and  J\Tf  being 
substituted  in  Equation  (704),  and  that  equation  solved  with  refer- 
ence to  P,  will  give  the  relation  in  question. 

§380. — If  the   power  P  act    in    the    direction  of  the  resistance   Q, 
then   will   cos  9  sac. I,    sin  9  =  0,    and     Equation     (704)     would,    after 
substituting     the     corresponding     values    of    iV  and    iV7,    transposing 
omitting   the    common    factor  d  Sj ,   and    supposing   p  =  p',   become 

PB=  Qr+f?(w+  Q  +  P)  4-  drK+IQ  -r.  •  ■  (707) 

And  omitting  the  terms  involving  the  friction  and  stiffness  of 
cordage, 

P_  _     r_ 

Q  ~  ~R% 

that  is,  the  power  is  to  the  resistance  as  the  radius  of  the  arbor 
is  to  that  of  the  wheel ;  which  relation  is  exactly  the  same  as 
that   of  the   common    lever. 

FIXED   PULLEY. 

§381.— -The    pulley  is   a    small  wheel    having  a  groove  in    its   cir- 
cumference  for    the    reception    of  a   rope,    to    one    end   of    which    the 


472 


ELEMENTfc     OF     ANALYTICAL    MECHANICS. 


power  P  is   applied,  and    to   the  other  the  resistance   Q.     The  pulley 
may  turn  either  upon   trunnions  or  aoout  an  axle,  supported  in  what 


is  called  a  block.  This  is  usually  a  solid  piece  of  wood,  through 
which  is  cut  an  opening  large  enough  to  receive  the  pulley,  and 
allow  it  to  turn  freely  between  its  cheeks.  Sometimes  the  block  is 
a  simple  framework  of  metal.  When  the  block  is  stationary,  the 
pulley  is  said  to  be  fixed.  The  principle  of  this  machine  is  obvi- 
ously the  same   as  that   of  the  wheel  and  axle. 

The  friction  between  the  rope  and  pulley  will  be  sufficient  to 
give  the    latter   motion. 

Making,  in  Equations  (705)  and  (706), 


we   have 


g  =  q  =p  =  $  I, 


N  =  i  y/{w  +  Q  +  P  cos  <p)2  -f  P2  sin2  ?  =  JST  .  .  (708) 


Making  R  =  r,  and  p  =  p',  in  Equation  (704),  and  substituting 
the  above  values  of  N  and  N\  we  have,  after  omitting  the  common 
factor  ds. 


>n 


PR-  QR-f'ty/{w+  #  +  Peos<p)2+P2siu2<p-.  d,  •  K**®-R=0.  •  (709) 

&  /I 


APPLICATIONS 


473 


Solving  this  equation  with  respect  to  P,  we  find  the  value  of 
the  latter  in  terms  of  the  different  sources  of*  resistance.  But  this 
direct  process  would  be  tedious ;  and  it  will  be  sufficient  in  all 
cases  of  practice  to  employ  an  approximate  value  for  P  under  the 
radical,  obtained  by  rirst  neglecting  the  terms  involving  friction  and 
stiffness  of  cordage. 

Thus,  dividing  by  R  and    transposing,  we  find 


K+IQ 


P  =  C  +^  R   vV  +  Q  +  P  cos  <p)2  -h  P2  sin2  <p  +  dt 


Now  /*'  •  -5-    is    usually  a    small  fraction  ;    an    erroneous    value  as. 
R 

ruined  for  P  under  the  radical,  will    involve    but   a   trifling   error    in 
lie    result.     We   may  therefore  write   Q  for  P  in    the    second    mem- 
per ;   and    neglecting    the  weight   of  the   pulley,  which    is   always    in- 
significant  in    comparison   to   Q,  we   have 


p  =  Q  [I  +  f .  ±^2(1  +  cos?)}  +  d,-^?®-  - 


%R 


hut 


1  -f-  cos  <p  =  2  cos2  J  <p  ; 


whence, 


J>=V(1  +2/'i.cosi9)  +  rf,.^j^ 


•  •  • 


(710) 


(711) 


in  which  <p  denotes  the  angle  A  M  2?,  which 
is  the  supplement  of  the  angle  A  0  B,  and  de- 
noting this  latter  angle  by  6,  we  have 


whence 


cos  I  9  =  sin  I  &  , 


p=Q(\+2f-^sml&)  +  d4 


K+IQ 
2R 


(712) 


If  the   arc    of    the    pulley,    enveloped    by    the   :*ope,    be    180°,    then 
will 


P=Q(l  +  2/\iL)-M, 


K  +  IQ 
2R 


(713) 


474 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


If  the  friction  and  stiffness  of  .ordage  be  so  small   as  to  justify  their 
omission,  then  will 

P  =  Q. 

That   is.  the   power   must    be    equal    tc  the   resistance,  and    the  only 
office  of  the  cord  or  rope  is  to  change  the  direction  of  the  power. 


MOVABLE    PULLEY. 

§  382. — In  the  fixed  pulley,  the  resultant  action  of  the  power  and 
resistance  is  thrown  upon  the  trunnion  boxes.  If  one  end  of  the 
rope  be  attached  to  a  fixed  hook  A, 
while  the  power  P  is  applied  to  the 
other,  and  the  pulley  is  left  free  to  roll 
along  the  rope,  the  resistance  W  to  be 
overcome  may  be  connected  with  its 
trunnion,  after  the  manner  of  the  figure ; 
the  pulley  is  then  said  to  be  movable, 
and  the  relation  between  the  power  and 
resistance  is  still  given  by  Eq.  (704,) 
in  which  the  principal  resistance  be- 
comes N  -f-  N\  and  the  tension  of  the 
rope  between  the  fixed  point  A,  and  the 
tangential  point  H,  becomes   Q. 

Making  in  Equation  (704),  R  =  r,  p  =  p',  and  W  =  N  +  Nr=  2  Ny 
we  have 


P  R-QR-ffW-d, 
dividing  by  R,  and  transposing 


2R 


R  =  0 


(714) 


2  R 


(715) 


Eliminating  Q  by  means  of  Equation  (708),  and  solving  the  resulting 
equation  with  respect  to  P,  the  value  of  the  power  will  be  known 
in  terms  of  the  resistances.  The  process  may  be  much  abridged  by 
limiting  the  solution  to  an  approximation,  which  will  be  found  suffi- 
cient in  practice. 


APPLICATIONS 


475 


Neglecting  the  weight  of  the  pulley,  which  is  always  insignificant 
in  comparison  with  P  or  Q,  and  making  Q  —  P,  which  would  be  the 
case  if  we  neglect  friction  and  stiffness  of  cordage,  Equation  (708), 
gives 


and  because 


JV*=  J  W=  iQ  y2(l  +  cos<p); 

1  -f  cos  (p  =  2  cos2  £  p  =  2  sin2  £  d, 
W  =  2  Q  .  sin  £  t)  ; 


or. 


G  = 


JF 


2  sin  ±6  ' 


which,  in  Equation  (715),  gives 


P-*fcrot>-i)>.*' 


iT+  /. 


IT 


2  sin  4-  4 


2i2 


.     (710) 


The    ouantity   of   work    is   found   by    multiplying    both    members   by 
R  $i ,  in   which  sx   is  the   arc  described  at   the   unit's  distance. 

If  the   arc   enveloped   by  the   rope    be    1 80°,   then  will   £  Q  =  90°, 
sin  \  &  =  1,  and 


P=  W  (l  +/>■■£)  +  <*,- 


2B 


(»«) 


If  the   friction    and   stiffness  of    cordage    be   neglected,    then    will. 
Equation  (716), 

W  =  2  P  sin  J  0, 

and  multiplying  by  P, 

R  W  =  P  .  2  R  .  sin  |  6  ; 


but 
xr  hence, 


2  P  sin  £  d  =  A  B ; 


P.  TT=  P  .  AB; 

that  is,  $e  power  is  to  the  resistance  as  the 
radius  of  the  pulley  is  to  the  chord  of  the  arc 
enveloped  by  the  rope. 


476 


ELEMENTS     OF     ANALYTICAL     MECHANICS 


§  383. — The  Muffle  is  a  collection  of  pulleys  in  two  separate 
«  blocks  or  frames.  One  of  these  blocks  is  attached  to  a  fixed  point 
A\  by  which  all  of  its  pulleys  become  Jixed, 
while  the  other  block  is  attached  to  the  resist- 
ance W7,  and  its  pulleys  thereby  made  mov- 
able. A  rope  is"  attached  at  one  end  to  a  hook 
h  at  the  extremity  of  the  fixed  block,  and  is 
passed  around  one  of  the  movable  pulleys, 
then  about  one  of  the  fixed  pulleys,  and  so  on, 
in  order,  till  the  rope  is  made  to  act  upon  each 
pulley  of  the  combination.  The  power  P  is 
applied  to  the  other  end  of  the  rope,  and  the 
pulleys  are  so  proportioned  that  the  parts  of 
the  rope  between  them,  when  stretched,  are 
parallel.  Now,  suppose  the  power  P  to  main- 
tain in  uniform  motion  the  point  of  applica- 
tion of  the  resistance  W;  denote  the  tension 
of  the  rope  between  the  hook  of  the  fixed 
block  and  the  point  where  it  comes  in  con- 
tact with  the  first  movable  pulley  by  tx;  the 
radius  of  this  pulley  by  Rx ;  that  of  its  eye 
by  r, ;  the  co-efficient  of  friction  on  the  axle 
by  f\  the  constant  and  co-efficient  of  the  stiff- 
ness of  cordage  by  K  and  /,  as  before;  then,  denoting  the  tension  of 
the  rope  between  the  last  point  of  contact  with  the  first  movable, 
and  first  point  of  contact  with  the  first  fixed  pulley,  by  £2,  the  quan- 
tity  of  work  of  the  tension   tt  will,   Equation   (652).   be 

tx  Rx  s,  =  tx  R,  ^  +  d4  — iL-l1  jfc,  *,  +  f  (t,  +  t,)  r,  s, ; 


V 


2R, 


in  which 


r  = 


f 


VTT7' 


dividing  by  *,, 

tH  /v, 


ft  *i  Va,  •  ygffi  •  *  +  /  ft  +  *) r*  -    118) 


applications.  477 

Again,  denoting  the  tension  of  that  part  of  the  rope  which  passes 
from  the  first  fixed  to  the  second  movable  pulley  by  /3,  the  radius 
of  the  first  fixed  pulley  by  i?2 ,  and  that  of  its  eye  by  r, ,  we  shall, 
in  like  manner,  have 

t.R,  =  m  +  dt  *£J*  22,  +  f  («,  -f  k)  r,.      .     (719) 

And  denoting  the  tensions,  in  order,  by  tA  and  tb ,  this  last  being 
equal   to   P,  we  shall  have 

t4R3  =  tzR3  -f  d,  *****  'Ri+f  (4  +  h)  r,.     .     (720) 

PP4  =  *4  P4  +  dt  ^~  &*  +  /'  (U  +  P)  r+     .     (721) 

so  that  we  finally  arrive  at  the  power  P,  through  the  tensions  which 
are  as  yet  unknown.  The  parts  of  the  rope  being  parallel,  and  the 
resistance  W  being  supported  by  their  tensions,  the  latter  may  ob- 
viously be  regarded  as  equal  in  intensity  to  the  components  of  W\ 
hence, 

UUU^^;    •    •    •    •    •    (722) 

which,  with  the  preceding,  gives  us  five  equations  for  the  determi- 
nation of  the  four  tensions  and  power  P.  This  would  involve  a 
tedious  process  of  elimination,  which  may  be  avoided  by  contenting 
ourselves  with  an  approximation  which  is  found,  in  practice,  to  be 
sufficient! v  accurate. 

If  the   friction    and  stiffness    be    supposed    zero,    for    the    moment* 
Equations  (718)  to  (721)  become 

^R%  =  hR*, 
t4Rz  =  /,i?,, 
PR<  =  tKR^ 

from  which   it   is   apparent,  dividing   out    the   radii    P, ,   ff8,  Rt1  4tu, 


478  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

that  t%  =  tXi  t%  =  tf8,  tA  =  /3,  P  =  t4-,  and  hence,  Equation  (7221 
becomes 

4  tx  =  W; 
whence, 

W 

the  denominator  4  being  the  whole  number  of  pulleys,  movable  and 
fixed.     Had   there   been   n   pulleys,  then  would 

W 

tx  = 

n 

With  this  approximate  value  of  tl1  we  resort  to  Equations  (718) 
to  (721),  and  find  the  values  of  £2,  ts.  t4,  &c.  Adding  all  these 
tensions  together,  we  shall  find  their  sum  to  be  greater  than  W, 
and  hence  we  infer  each  of  them  to  be  too  large.  If  we  now 
suppose  the  true  tensions  to  be  proportional  to  those  just  found, 
and  whose  sum  is  Wx  >  W,  we  may  find  the  true  tension  corre- 
sponding to  any  erroneous  tension,  as  tx ,  by  the  following  propor- 
tion, viz. : 

W 

rr  j    .     rr     .  .    tx    .  i, , 

or,  which   is  the  same  thing,  multiply  each  of  the    tensions  found  by 

W 

the    constant  ratio  — >   the    product  will    be    the    true    tensions,  very 

nearly.  The  value  of  t4  thus  found,  substituted  in  Equation  (721), 
will   give   that   of  P. 

Example. — Let  the  radii  Rx ,  /?9,  Rz  and  i24,  be  respectively 
0.26,  0,39,  0,52,  0,65  feet  ;  the  radii  r,  =  r2  =  r3  =  r4  of  the 
eyes  =  0,06  feet ;  the  diameter  of  the  rope,  which  is  white  and 
dry,  0,79  inches,  of  which  the  constant  and  co-efficient  of  rigidity 
are,  respectively,  K  =  1,6097  and  /  =  0,0319501  ;  and  suppose  the 
pulley  of  brass,  and  its  axle  of  wrought  iron,  of  which  the  co-efficient 
/  =  0,09,  and   the   resistance    W  a  weight  of  2400  pounds. 

Without  friction    and    stiffness  of  cordage, 

2400  '*»• 

tx  =  ——  =  600. 


APPLICATIONS 


479 


Dividing  Equation  (718)  by  22,,  it   becomes,  since  dt  =  1, 

Substituting  the  value  of  22, ,  and  the  above  value  of  Jj ,  and  regard, 
ng    in    the   last   term    t2  as    equal    to  £, ,  which  we  may  do,  because 

of  the    small  co-efficient  -^-  /',   we  find 

22, 


U=  \  + 


600 


1,6097  +  0,0319501   x  600 


2  x  (0,26) 


f   =  623,39. 


+  jjljj-  X  0,09  x  (600  +  600) 


Again,  dividing    Equation    (719)    by  H.^  and    substituting   this    value 
of  t.2  and   that  of  22,,  we  find 

lbs.  / 

t3  =  673,59. 

Dividing  Equation  (720)  by  fiz,  and  substituting  this  value  <f  /3,  H 
well   as    that   of  B^,  there  will   result 

lbs. 

t4  =  709,82 ; 


whence, 


JPi  =  *x  +  *t  +  *  +  «4  =  < 


600 
-f  628,39 
+  673,59 
L  +  709,82 


.  =  2611  #> 


and 


2400 


=  0,919 ; 


2611,80 

which  will  give  for  the   true    values  of 

tx  =  0,919  x  600  -  551,400 

t2  =z  0,919  x  628,39  =  577,490 

t3  =  0,919  X  673,59  as  619,029 

f4  =  0,919  x  709,82  =  652,324 


2400,243 


480 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


The  above  value  for  t4  =  05*2,324,  in   Equation  (^21),  will  give,  aflei 
dividing  by  i?4,  and  substituting  its  numerical  value. 


P  =  < 


+ 


652,324 

1,0097  +  0.03195  x  652,324 


2  x  0,05 


0,06 


f  ~g  X  0,09  x  (652,324  +  P)  \ 
tnd  making  in  the  last  factor  P  =  l4  =  '652,324,  we  find 

lbs.  lbs.  lbs.  lbs. 

P  =  652,324  -f  17.270  +  10,831  ==  680,425. 

Thus,  without  friction  or  stiffness  of  cordage,  the  intensity  of  P  would 
be  600  lbs. ;  with  both  oi  these  causes  of  resistance,  which  cannot  be 
..voided  in  practice,  it  becomes  680,425  lbs.,  making  a  difference  of 
S0,425  lbs.,  or  nearly  one-seventh ;  and  as  the  quantity  of  work  of 
the  power  is  proportional  to  its  intensity,  we  see  that  to  overcome 
friction  and  stiffness  of  rope,  in  the  example  before  us,  the  motor 
must  expend  nearly  a  seventh  more  work  man  if  these  sources  ot 
resistance  did  not  exist. 


THE    WEDGE. 


§  384. — The  wedge   is   usually   employed    in    the   operation   of  cut- 
ting,   splitting,  or  separating.      It  consists 
of  an  acute  right  triangular  prism  A  B  C. 
The   acute  dihedral   angle  A  Cb  is  called 
the   edge ;     the     opposite    plane    face    A  b 
the    lack-,    and    the    planes    Ac   and    Cb, 
which    terminate   in    the   edge,    the  faces. 
The    more    common     application    of    the 
wedge   consists   in   driving   it,  by  a   blow 
upon    its   back,    into  any  substance   which 
we  wish   to   split  or  divide  into  parts,  in 
such    mannei     that    after   each   advance   it 
shall   be    supported    against   the    faces   of 
♦he    opening    till    the    work    is    accomplished. 


APPLICATIONS. 


481 


§  385. — The  blow  by  which  the  wedge  is  driven  forward  will  be 
supposed  perpendicular  to  its  back,  for  if  it  were  oblique,  it  would 
only  tend  to  impart  a  rotary  motion,  and  give  rise  to  complications 
which  it  would  be  unprofitable  to  consider :  and  to  make  the  case 
conform  still  further  to  practice,  we  will  suppose  the  wedge  to  be 
isosceles. 

The  wedge  ACB  being  inserted  in  the  opening  a  hb,  and  in  col. 
tact  with  its  jaws  at  a  and  &,  we  know 
that  the  resistance  of  the  latter  will 
be  perpendicular  to  the  faces  of  the 
wedge.  Through  the  points  a  and  6 
draw  the  lines  a  q  and  b p  normal  to 
the  faces  A  C  'and  B  C ;  from  their 
point  of  intersection  0  lay  off  the 
distances  Oq  and  Op  equal,  respec- 
tively, to  the  resistances  at  a  and  b. 
Denote  the  first  by  Q,  and  the  second 
Dy  P.  Completing  the  parallelogram 
Oqmp,  0  m  will  represent  the  re- 
sultant of  the  resistances  Q  and  P. 
Denote  this  resultant  by  R',  and  the 
angle  A  C  B  of  the  wedge  by  $,  which, 
in    the    quadrilateral    a  0  b  C,   will    be 

<?qual  to  the  supplement  of  the  angle  a  0  b  —  p  0  q,  the  angle  made 
by  the  directions  of  Q  and  P.  From  the  parallelogram  cf  forces, 
we  have, 

Rfi-  p2+  Q2  +  2P  QcospOg  =  P2  +  Q2  -2Ptf  o«l; 


or. 


R'  =  y3*  +  Q*  -  2  P  Q  cos  6. 

The  resistance  Q  will  produce  a  friction  on  the.  face  A  C  equal 
to  fQ,  and  the  resistance  P  will  produce  on  the  face  B  C  the  fric- 
tion f  P '.  these  act  in  the  directions  of  the  faces  of  the  wedge.. 
Produce  them  till  they  meet  in  C\  and  lay  off  the  distances  C  q'  and 
Cpf  to    represent    their    intensities,    and    complete    the   parallelogram 


482  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

Cq'  0'pf\  CO' will  represent  the  resultant  of  the  frictions.  P^not* 
this  by  It",  and  we  have,  from  the  parallelogram  of  forces, 

B"2  =  p  if  -f/2P2  +  2/2  P  Q  cos  d ; 
or, 

R"  =  /y/P2  +  Q2  +  2  P  Q  cos  d. 

The  wedge  being  isosceles,  the  resistances  P  and  Q  will  be  equal, 
their  directions  being  normal  to  the  faces  will  intersect  on  the  line 
CD,  which  bisects  the  angle  (7  =  4,  and  their  resultant  will  coin- 
cide  with  this  line.  In  like  manner  the  frictions  will  be  equal,  and 
their  resultant  will  coincide  with  the  same  line.  Making  Q  and  P 
equal,  we  have,  from  the  above  equations, 

R'  =     P  y/2{\  —  cosT), 

R"  =  fP  y/2  (1  +  cosd). 

But, 

1  —  cos  6  =  2  sin2  \  &, 

» 

1  4-  cos  &  =  2  cos2  i  &  ; 

whence  we  obtain,  by   substituting  and  reducing, 

R'  ■=■  2  P.  sin  i  0, 
R"  =  2/.  P.  cos  }  a  ; 


and  further, 


therefore 


sm  *  ■  =  *  ~ACT 

cos  *    =    ITc  ; 

A  B 


R'  =     P- 


#"  =  2/-P 


^4  C" 
CZ) 


^4  C 


Denote  by  F  the   intensity  of  the  blow  on    the    back  of  t*ie  wedge. 
.If  this   blow   be  just    sufficient    to  produce   an    equilibrium    bordering 


APPLICATIONS.  433 

on  motion  forward,  call  it  F' ;  the  friction  will  oppose  it,  and  we 
must   havu 

F'  =  R'  +R"  =  p.jJL  +  Zf.p.-£.i2J.    .     .     .     (725) 

If,  or:  the  contrary,  the  blow  be  just  sufficient  tc  prevent  the  wedge 
from  flying  back,  call  it  F"  ;  the  friction  will  aid  it,  and  we  must 
have. 

F"-P.— 2f.p.— —    ....     (724) 

AC  J  AC  V       ' 

• 

The  wedge  will  not  move  under  the  action  of  any  force  wrhose  inten- 
sity is  between  F'  and  F".  Any  force  less  than  F",  will  allow  it 
to  fly  back  ;  any  force  greater  than  F',  will  drive  it  forward.  The 
range  through  which  the  force  may  vary  without  producing  motion, 
is  obviously, 

F'-F"  =  ifP-^-    .....     (725) 

which  becomes  greater  and  greater,  in  proportion  as  C  D  and  A  0 
become  more  nearly  equal ;  that  is  to  say,  in  proportion  as  the 
wedges  becomes  more  and  more  acute. 

The  ordinary  mode  of  employing  the  wedge  requires  that  it  shall 
retain  of  itself  whatever  position  it  may  be  driven  to.  This  makes 
it  necessary  that  F"  should  be  zero  or  negative,  Eq.  (724),  whence 

PA±  =  if.P.l^.,  W?ii<8/.,P  CD 


A  C   ~     J  AC"  A  C    ^     J  AC' 

or,  omitting  the  common  factors  and  dividing  both  members  of  the 
equation   and   inequality  by  2  C  J), 

$A  B  \AB 

-CD=f\°T        CD<fi 

1  A  B 

but     ?  is    the  tangent  of  the    angle  A  C  D ;  hence  we  conclude, 

Kj  u 

that  the  wedge  will  retain  its  place  when  its  semi-angle  does  not 
exceed  that  whose  tangent  is  the  co-efficient  of  frictit  n  between  the 
surface  of  the  wedge  and  the  surface  of  the  opening  which  it  ii 
intended  to  enlarge. 


484  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Resuming  Eq.  (724),  and  supposing  the  last  term  of  the  second 
member  greater  than  the  first  term,  F"  becomes  negative,  and  will 
represent  the  intensity  of  the  force  necessary  to  withdraw  the  wedge ; 
which  will  obviously  be  the  greatest  possible  when  A  B  is  the  least 
possible.  This  explains  why  it  is  that  nails  retain  with  such  pert! 
i  acity  their  places  when  driven  into  wood,  &c. 


THE   SCREW. 

§  386. — The  Screw,  regarded  as  a  mechanical  power,  is  a  device  by 
which  the  principles  of  the  inclined  plane  are  so  applied  as  to  pro- 
duce considerable  pressures  with  great  steadiness  and  regularity  of 
motion. 

To  form  an  idea  of  the  figure  of  a  screw  and  its  mode  of  action, 
conceive  a  right  cylinder,  a  k,  with  circular  base,  and  a  rectangle,  or 
other  plane  figure,  abem,  having  one  of  its  sides 

a  b   coincident  with  a  surface  element,  while  its  s*r** ^N 

plane  passes   through   the  axis  of  this  cylinder.  K^~~ — ■ iN 

Next,   suppose   the   plane  of  the   generatrix    to  L^^:::::::::^^ 

rotate  uniformly  about  the  axis,  and  the  gener-  ^^^     C 

atrix  itself  to  move  also  uniformly  in  the  direc- 
tion of  that'  line  ;    and  let  this  twofold    motion  0f 
of  rotation  and  of  translation    be  so  regulated,           m\ 
that   in  one  entire   revolution  of  the  plane,  the  il~~      «* 
generatrix    shall    progress    in    the    direction    of 

the  axis  over  a  distance  greater  than  the  side  a  b,  which  is  in  the 
surface  of  the  cylinder.  The  generatrix  will  thus  generate  a  pro- 
jecting and  winding  solid  called  a  fillet,  leaving  between  its  turns 
a  groove  called  the  channel.  Each  point  as  m  in  the  perimeter 
of  the  generatrix,  will  generate  a  curve  called  a  helix,  and  it  is 
obvious,  from  what  has  been  said,  that  every  helix  will  enjoy  this 
property,  viz.  :  any  one  of  its  points  as  m,  being  taken  as  an  origin 
of  reference,  as  well  for  the  curve  itself  as  for  its  projection  on  a 
plane  through  this  point  and  at  right  angles  to  the  axis,  the  distances 
dr  m\d"  m",  &cM  of  the  several    points  of  the  helix  from    this  plane, 


APPLICATIONS. 


485 


are  respectively  proportioned  to  the  circular  arcs  md\  md",  &c. 
into  which  the  portions  mm',  mm",  &c,  of  the  helix,  between  the 
origin   and    these    points,    are   projected. 

The  solid  cylinder  about  which  the  fillet  is  wound,  is  called 
the  newel  of  the  screw ;  the  distance  m  m'",  between  the  consecu- 
ti\3  turns  of  the  same  helix,  estimated  in  the  direction  of  the  axis, 
is    called    the    helical   interval. 

The  fillet  is  often  generated  by  the  motion  of  a  triangle  with 
one  of  its  sides  coincident  with  ab\  and  as  the  discussion  will  be 
more  general  by  considering  this  mode  of  generation,  we  shall  adopt 
it.  The  surfaces  of  the  fillet,  which  are  generated  by  the  inclined 
faces   of    the    triangle,  are    each   made   up   of    an    infinite   number  of 

helices,    all    of    which    have    the   same    interval,    though    the    helices 

« 

themselves  are  at  different  distances  from  the  axis,  and  have  different 
inclinations   to    that  line. 

The  inclination  of  the  different  helices  to  the  axis  of  the  screw, 
increases  from  the  newel  to  the  exterior  surface  of  the  fillet, 
the  same  helix  preserving  its 
inclination  unchanged  throughout. 
The  screw  is  received  into  a  hole 
in  a  solid  piece  B  of  metal  or 
wood,  called  a  nut  or  burr.  The 
surface  of  the  hole  through  the 
nut  is  furnished  with  a  winding 
fillet  of  the  same  shape  and  size 
as  the  channel  of  the  screw,  so 
that  the  surfaces  of  the  screw  and 
nut  are  brought  into  accurate  con- 
tact. 

From  this  arrangement  it  is 
obvious  that  when  the  nut  is  sta- 
tionary, and  a  rotary  motion  is" 
communicated  to  the  screw,  the 
latter  will  move  in  the  direction 
of    its    axis ;     also,    when    the    screw    is   stationary    and    the   nut    is 

turned,    the   nut    must   also    move   in   the    direction    of  the   axis.      In 

30 


486 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


the  first  case,  one  entire  revolution  of  the  screw  will  carry  it  lon- 
gitudinally through  a  distance  equal  to  the  helical  interval,  and  any 
fractional  portion  of  an  entire  revolution  will  carry  it  through  a  pro- 
portional distance ;  the  same  of  the  nut,  when  the  latter  is  mova- 
ble and  the  screw  stationary.  The  resistance  Q  is  applied  either  to 
the  head  of  the  screw,  or  to  the  nut,  depending  upon  which  is,  the 
movable  element ;  in  either  case  it  acts  in  the  direction  D  C  of 
the  axis.  The  power  P  is  applied  at  the  extremity  of  a  bar  GH 
connected  with  the  screw  or  nut,  and  acts  in  a  plane  at  right 
angles  to    the    axis    of  the    screw. 

From    the    description    of  the    screw  and    its   mode    of  generation, 
we    may   find    the    equation    of   its    fillet   or   helicoidal    surface.      For 
this  purpose,  take  the  axis  z  to  coincide  with  the   axis  of  the  newel, 
and   the   initial    position  of  the   generatrix  in    the   plane  yz.     Make 
*   =  any    definite   portion    C  C 

of  an  assumed  helix  ; 
<p  =  the    angle    Y A  t,  'through 
which    the    rotating    plane 
has  turned  during  the  gene- 
ration of  s  ; 
r  —.  the   di.>tance    CD    of    this 

helix  from    the   axis  z ; 
a  =  the-  angle  which  this    helix 
makes  with  the  plane  xy\ 
£  =  the  angle   C  BD  which  the 
generatrix  of  the  helicoidal 
surface     makes     with     the 
axis  z  ; 
y  =  the  co-ordinate  AB  of  the 
point  in  which   the  genera- 
trix, in  its  initial    position,  intersects   the   axis  z. 
Then,  for  any  point   as   C  of  the  generatrix  in  its  initial  position, 
we    have 

z  —  AD  —  AB  -f  BD  =  y  +  r.  cotan  €, 

and    for    any  subsequent  position,  as   C  B\ 

z  =  y  -f-  r  .  cotan  §  +  r  .  9  .  tan  a,  •     •     .     •     (726) 


APPLICATIONS.  487 

which  is  the  equation  sought,  and  in  which  a  and  r  are  constant 
for   the    same   helix,  and  variable  from  one  helix    to  anothei 

The   power   P   acts  in   a   direction    perpendici  lar    to   the    axis  uf 
the  newel.     Denote    by  /  its   lever  arm  ;  its  virtual    moment  will  be 

',  Pldy. 

The  resistance  Q  acts  in  the  direction  of  the  axis  of  the  newel ; 
its    virtual    moment  will    be 

Qdz. 

The  friction  acts  in  the  direction  of  the  helicoidal  surface  and  paral- 
lel to  the  helices.  Conceive  it  to  be  concentrated  upon  a  mean 
helix,  of  which  the  distance  from  the  newel  axis  is  r,  and  length  s : 
denote  the  normal  pressure  by  JV,  and  co-efficient  of  friction  by  f. 
The  virtual  moment  of  friction  will  be 

,  *  f.N.d*\ 

and  Equation  (645), 

Pldcp  -  Qdz  -j\N.ds  =  0 (727) 

But  the  displacement  must  satisfy  Equation  (726),  or,  as  in  §  213, 
the  condition, 

L  =  z  —  r  .  (p  .  tan  a  —  r  .  cotan  €  —  y  =  0;       .     (728) 

and  also, 

r  =  constant (729) 

Differentiating,  we  have, 

dz  —  cotan  £  .  d  r  —  r  tan  a  dcp  =  0, 

dr  =  0. 

Multiplying  the  first  by  X,  the  second  by  X',  adding  to  Equation 
(727),  and  eliminating  d  s  by  the  relation 

d s  =  r  .  dcp  .  cos  a  +  dz  .  sin  a,    .  .     ,      (730) 

we  find, 
tPl—  f.N.ooaa  .r  -  \tena.r)df  +  (X  -  Q  - /.  JV*.  sin  a)  dz  +(A'-  Xcotan€)(/r  -  0 


488  ELEMENTS     OF     ANALYTICAL     MECHANICS 

and,  from   the  principle  of  indeterminate  co -efficients, 

PI  —  f .  N .  cos  a  .  r  —  X  .  tan  ct  .  r  =  0;     .     .  (731) 

Q  +fN.  sin  a  -  X  =  0; (732) 

X'  —  X  cotan  *  -  0 (732)' 

The    variables   d  z,  d  r,  andrd<p,  are    rectangular;     whence,    Equation 
(331), 


^  - x  v  (-,77)  +  (77)  +  W  - x  Vl  +  ta"  a  +  C0t8",e- 

Substituting  this  in  Equations  (731)  and  (732),  and  eliminating  /., 
there  will  result 


„         _     r      tan  a   -f-  f .  cos  a  .  <J  \   4-  tan2  a  -f-  cotan'2? 

P  =   Q'j- V  = .     (733) 

L  1  —  /.  sin  a  .  yl  -f-  tan2  a  -(-  cotan2  € 

Substituting    the    value    of   X    from    Equation    (732),  in    Equatior. 
(73  J)',  we  find, 

->  ,        ^  •     cotan  § 

V  =  G ; -=  ;    .     (734) 

1   —  /.  sin  a  y^l   -f  tan2  a  -{-  cotan2  § 

in  whicrt  X'  is,  §  217,  the  value    of  the    force    acting    in    the  direction 
of  r. 

§  387.— If  the  fillet    be    nectangular,  §  —  90°,  cotan  §  =  0,  and 

_         _     r     tan  a  +  /.  cos  a  .  -i/l  4-  tan2  a  ,.       v 

P  =   Q  . ^ . V      "  .      .     (735) 

*  1  —  /  .  sin   a  .  yl  -|-  tan2  a 

ftni 

X'  r=  0. 

§  388. — If  we  neglect  the  friction,  /  =  0  ;  and 

PI  =  $  .  r  .  tan  a, 
multiplying  both  members  by  2  at, 

P  .  2  ^r  /  =  #  .  2  o  .  tan  a (736) 

That    is,   the  power  is  to  the  resistance   as    tlie  helical   interval   is    to 
the  circumference  described  by   the  end  of  the  level  arm   of  the  power. 


APPLICATIONS. 


489 


TUMI'S. 


§  389. — Any  machine  used  for  raising  liquids  from  one  level 
to  a  higher,  in  which  the  agency  of  atmospheric  pressure  is  employed, 
is  called  a  Pump.  There  are  various  kinds  of  pumps ;  the  more 
common  are  the  bucking,  forcing,  and  lifting  pumps. 

§  390. — The  Sacking-Pump  consists  of  a  cylindrical  body  or  barrel 
B,  from  the  lower  end  of  which  a  tube  D,  called  the  sucking-pipe, 
descends  into  the  water  contained  in  a  reservoir  or  well.  In  the 
interior  of  the  barrel  is  a  movable  piston  C,  surrounded  with  leather 
to  make  it  water-tight,  yet  ca- 
pable of  moving  up  and  down 
freely.  The  piston  is  perforated 
in  the  direction  of  the  bore  of 
the  barrel,  and  the  orifice  is 
covered  by  a  valve  F  called 
the  piston-valve,  which  opens  up- 
ward  ;  a  similar  valve  E,  called 
the  sleeping-valve,  at  the  bottom 
of  the  barrel,  covers  the  upper 
end  of  the  sucking-pipe.  Above 
the  highest  point  ever  occupied 
by  the  piston,  a  discharge-pipe 
P  is  inserted  into  the  barrel  ; 
the  piston  is  worked  by  means 
of  a  lever  H,  or  other  contriv- 
ance, attached    to  the  piston-rod 

G.  The  distance  A  A',  between  the  highest  and  lowest  points  of  the 
piston,  is  called  the  play.  To  explain  the  action  of  this  pump,  let 
the  piston  be  at  its  lowest  point  A,  the  valves  E  and  F  closed  by 
their  own  weight,  and  the  air  within  the  pump  of  the  same  density 
and  elastic  force  as  that  on  the  exterior.  The  water  of  the  reservoir 
will  stand  at  the  same  level  L  L  both  within  and  without  the 
sucking-pipe.  Now  suppose  the  piston  raised  to  its  highest  point  A', 
the    air    contained    in    the    barrel     and    sucking-pipe    wijl    tend    by   ita 


D 


490  ELEMENTS-OF     ANALYTICAL    MECHANICS 

elastic  force  lo  occupy  the  space  which  the  piston  leaves  void,  the 
valve  E  will,  therefore,  be  forced  open,  and  air  will  pass  from  the 
pipe  to  the  barrel,  its  elasticity  diminishing  in  proportion  as  it  fills 
i  larger  space.  It  will,  therefore,  exert  a  less  pressure  on  the 
water  below  it  in  the  sucking-pipe  than  the  exterior  air  does  on  that 
in  the  reservoir,  and  the  excess  of  pressure  on  the  part  of  the 
exterior  air,  will  force  the  water  up  the  pipe  till  the  weight  of  the 
suspended  column,  increased  by  the  elastic  force  of  the  internal  air, 
becomes  equal  to  the  pressure  of  the  exterior  air.  When  this  takes 
place,  the  valve  E  will  close  of  its  own  weight;  and  if  the  piston 
be  depressed,  the  air  contained  between  it  and  this  valve,  having 
its  density  augmented  as  the  piston  is  lowered,  wall  at  length  have 
its  elasticity  greater  than  that  of  the  exterior  air ;  this  excess  of 
elasticity  will  force  open  the  valve  F,  and  air  enough  will  escape 
to  reduce  what  is  left  to  the  same  density  as  that  of  the  exterior 
air.  The  valve  F  will  then  fall  of  its  own  weight;  and  if  the 
piston  Vie  again  elevated,  the  water  will  rise  still  higher,  for  the 
same  reason  as  before.  This  operation  of  raising  and  depressing 
the  piston  being  repeated  a  few  times,  the  water  will  at  length  entei 
the  barrel,  through  the  valve  F,  and  be  delivered  from  the  dis- 
charge-pipe P.  The  valves  E  and  F,  closing  after  the  water  has 
passed  them,  the  latter  is  prevented  from  returning,  and  a  cylinder 
of  water  equal  to  that  through  which  the  piston  is  raised,  will,  at 
each  upward  motion,  be  forced  out,  provided  the  discharge-pipe  is 
large  enough.  As  the  ascent  of  the  water  to  the  piston  is  pro- 
duced by  the  difference  of  pressure  of  the  internal  and  external  air, 
it  is  plain  that  the  lowest  point  to  which  the  piston  may  reach, 
should  never  have  a  greater  altitude  above  the  water  in  the  reser 
voir  than  that  of  the  column  of  this  fluid  which  the  atmospheric 
pressure    may    support,    ir    vacuo,    at    the    place. 

§391. — It  will  readily  appear  that  the  rise  of  water,  during 
each  ascent  of  the  piston  after  the  first,  depends  upon  the  expulsion 
of  air  through  the  piston-valve  in  its  previous  descent.  But  air  can 
only  issue  through  this  valve  wrhen  the  air  below  it  has  a  greater 
density    and    therefore   greater    elasticity    than   the    external    air ;   am1 


APPLICATIONS, 


491 


if  the  piston  may  not  descend  low  enough,  for  want  of  sufficient 
play,  to  produce  this  degree  of  compression,  the  water  must  cease 
to  rise,  and  the  working  of  the  piston  can  have  no  other  effect  ib*r 
alternately  to  compress  and  dilate  the  same 
air  between  it  and  the  surface  of  the  water. 
To  ascertain,  therefore,  the  relation  which  the 
play  of  the  piston  should  bear  to  the  otjier 
dimensions,  in  order  to  make  the  pump  effec- 
tive, suppose  the  water  to  have  reached  a  sta- 
tionary level  X,  at  some  one  ascent  of  the 
piston  to  its  highest  point  A\  and  that,  in  its 
subsequent  descent,  the  piston-valve  will  not 
open,  but  the  air  below  it  will  be  compressed 
only  to  the  same  density  with  the  external  air 
when  the  piston  reaches  its  lowest  point  A. 
The  piston  may  be  worked  up  and  down  in- 
definitely, within  these  limits  for  the  play, 
without    moving,  the  water.      Denote    the   play 

of  tne  piston  by  a ;  the  greatest  height  to  which  the  piston  may  be 
raised  above  the  level  of  the  water  in  the  reservoir,  by  b.  which  mav 
also  be  regarded  as  the  altitude  of  the  discharge  pipe ;  the  elevation 
of  the  point  X,  at  which«  the  water  stops,  above  the  water  in  the 
reservoir,  by  x  ;  the  cross-section  of  the  interior  of  the  barrel  by  B 
The  volume  of  the  air  between  the  level  X  and  A  will  be 


B  x  (b  —  x  —  a) ; 

the  volume  of  this  same  air,  when  the  piston  is  raised  tc  A',  pro- 
vided the  water   does  not   move,  will    be 

B  (b  -  x). 

Represent  by  h  the  greatest  height  to  which  water  may  be  supported 
in  vacuo  at  the  place.  The  weight  of  the  column  of  water  which 
the  elastic  force  of  the  air,  when  occupying  the  space  between  the 
limits  X  and  A,  will  support  in  a  tube,  with  a  bore  equal  to  that 
of  the  barrel    is  measure!    by 

Bh.ff.  D; 


492  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

in  which  D  is  the  density  of  the  water,  and  g  the  fcrce  of  gravity. 
The  weight  of  the  column  which  the  elastic  force  of  th  3  same  «*ir 
will    support,  when    expanded   between    the    limits  X  and  A\  will   ue 

Bh'.g.D\ 

in  which  /*'  denotes   the    height  of  this  new  column.     But,  fr.  m  Ms* 

riotte's   law,  we  have 

♦ 

B  (b  - ■  x  -  a)  :  B(b  —  x)   :  :   B h' g  D  :  Bhg  D\ 

whence, 

b  —  x  —  a 


h'  =  h> 


b  —  x 


But  there  is  an  equilibrium  between  the  pressure  of  the  external 
air  and  thr«t  of  the  rarefied  air  between  the  limits  X  and  Af,  when 
the  latter  is  increased  by  the  weight  of  the  column  of  water  whose 
altitude  is  x.^   Whence,  omitting  the  common  factors  B,  D  and  g, 

b  —  x  —  a 

x  -\-  h    =  z  -\-  h-  — ; —  h  ; 

6  —  x 

or,  clearing    the  fraction  and   solving    the    equation  in    reference  to  #, 
we  find  • 


x  -  £6  ±  $  ^b2  -  4  ah. (737) 

When  x  has  a  real  value,  the  water  will  cease  to  rise,  but  x 
will  be  real  as  long  as  b2  is  greater  than  4  a  h.  If,  on  the  con- 
trary, 4  a  h  is  greater  than  b2,  the  value  of  x  will  be  imaginary,  and 
the  water  cannot  cease  to  rise,  and  the  pump  will  always  be  effective 
when    its   dimensions    satisfy  this   condition,  viz.  : — 

4  a  h  >  62, 

or, 

b2 
a  > 


4h 


that  is  to  say,  the  play  of  the  -piston  must  be  greater  titan  the  square 
of  the  altitude  of  the  upper  limit  of  the  play  of  the  piston  above 
the  surface  of  the  water  in  the  reservoir,  divided  by  four  ti'ues  the 
height    to   which    the  atmospheric  pressure  at  tlie  place,    where   the  pump 


APPLICATIONS. 


493 


is  used,  will  support  water  in  vacuo.  This  last  height  is  easily  found 
by  means  of  the  barometer.  We  have  but  to  notice  the  altitude 
of  the  barometer  at  the  place,  and  multiply  its  column,  reduced  to 
feet,  by  13J,  this  being  the  specific  gravity  of  mercury  referred  to 
water  as  a  standard,  and  the  product  will  give  the  value  of  .A  in 
feet. 

Example. — Required  the  least  play  of  the  piston  in  a  sucking- 
pump  intended  to  raise  water  through  a  height  of  13  feet,  at  a 
place  where   the   barometer  stands   at  .  28   inches.  j 


Here 

Barometer, 


b  =  13,     and     b2  =  169. 


*n. 


2S 

—  =  2,333  feet. 

12    . 


ft. 


h  =  2,333  X  13,5  =  31,5  feet. 


Play 


b2 

=  «  >77 


169 


ft 


4A        4  x  31,5 


=  1,341  +  ;     I 


7* 


V  :, 


M 


that   is,  the   play  of  the   piston   must   be  greater   than    one  and   on 
third  of  a   foot. 

i 

§  392. — The  quantity  of  work  performed  by 
the  motor  during  the  delivery  of  water  through 
the  discharge-pipe,  is  easily  computed.  Sup- 
pose the  piston  to  have  .any  position,  as  M, 
and  to  be  moving  upward,  the  water  being 
at  the  level  LL  in  the  reservoir,  and  at  P 
in  the  pump.  The  pressure  upon  the  upper 
surface  of  the  piston  will  be  equal  to  the 
entire  atmospheric  pressure  denoted  by  A, 
increased  bv  the  weight  of  the  column  of 
water  MP',  whose1  height  is  e',  and  whose 
base  is  the  area  B  of  the  piston  \  that  is,  the 
pressure   upon    the   top  of  the   piston  will  be 

A  +  Bc'gD, 

in  which  g  and   D  are  the  force  of  gravity  and  density  of  the  water, 
respectively       Again,    the   pressure    upon    the    undei    surface  of   the 


x  JV 


494  ELEMENTS    Of    ANALYTICAL    MECHANICS. 

piston  is  equal  to  the  atmospheric  pressure  A,  transmitted  through 
the  water  in  the  reservoir  and  up  the  suspended  column,  diminished 
by  the  weight  of  the  column  of  water  JVM  below  the  piston,  and 
jf  which  the  base  is  B  and  altitude  c ;  that  is,  the  pressure  from 
below  will   be 

A  —  BcgD, 
and   the   difference  of  these   pressures  will   be 

A  +  B  c'  g  B  —  (A  -  B  eg  D)  =  B g  D  (c  +  c') ; 
but,  employing  the   notation   of  the   sucking-pump  just   described, 

C  +  C'   =  ft; 

whence,  the  foregoing   expression   becomes 

Bb.g.B-, 

which  is  obviously  the  weight  of  a  column  of  the  fluid  whose  base 
is  the  area  of  the  piston  and  altitude  the  height  of  the  discharge-pipe 
above  the  level  of  the  "water  in  the  reservoir.  And  adding  to  this 
the  effort  necessary  to  overcome  the  friction  of  the  parts  of  the  pump 
when  in  motion,  denoted  by  (p,  we  shall  have  the  resistance  which  the 
force  F,  applied  to  the  piston-rod,  must  overcome  to  produce  an} 
useful  effect ;  that  is, 

F  =  BbgD  +  <p. 

Denote  the  play  of  the  piston  by  p,  and  the  number  of  its  double 
strokes,  from  the  beginning  of  the  flow  through  the  discharge-pipe 
till  any  quantity  Q  is  delivered,  by  n ;  the  quantity  of  work  will,  by 
omitting  the  effort  necessary  to  depress  the  piston,  be 

Fnp  =  np  [Bb  .  g  D  +  <p]; 

©r  estimating  the  volume  in  cubic  feet,  in  which  case  p  and  b  must 
be  expressed  in  linear  feet  and  B  in  square  feet,  and  substituting  for 
g  D  its  value  62,5  pounds,  we  finally  have  for  the  quantity  of  work 
rscessary  to  deliver  a  number  of  cubic  feet  of  water   Q  =  Bnpi 

Fnp  =  np[G2,5  .  Bb  +  <p] ;      ....     (738) 

in  which  <p   must   be  expressed   in   pounds,  and   may  be  determined 


APPLICATIONS 


495 


either  by   experiment  in   each  particular   pump,  or   computed   by  the 
rules  already  given. 

It  is  apparent  that  the  action  of  the  sucking-pump  must  be  very 
irregular,  and  that  it  is  only  during  the  ascent  of  the  piston  that  it 
produces  any  useful  effect;  during  the  descent  of  the  piston,  the  force 
is  scarcely  exerted  at  all,  not  more  than  is  necessary  to  overcome 
the  friction. 


§  393. — The  Lifting-Pump  does  not  differ  much  from  the  sucking 
pump  just  described,  except  that  the  barrel  and  sleeping-valve  E  are 
placed  at  the  bottom  of  the  pipe,  and  some  distance  below  the  sur 
face  of  the  water  L  L  in  the  reservoir  ;  the 
piston  may  or  may  not  be  below  this 
same  surface  when  at  the  lowest  point  of 
its  play.  The  piston  and  sleeping-valves 
open  upward.  Supposing  the  piston  at  its 
lowest  point,  it  will,  when  raised,  lift  the 
column  of  water  above  it,  and  the  pres- 
sure of  the  external  air,  together  with  the 
head  of  fluid  in  the  reservoir  above  the 
level  of  the  sleeping-valve,  will  force  the 
latter  open ;  the  water  will  flow  into  the 
barrel  and  follow  the  piston.  When  the 
piston  reaches  the  upper  limit  of  its  play, 
the  sleeping-valve  will  close  and  prevent 
the    return    of    the    water    above   it.      The 

piston  being  depressed,  its  valves  F  will  open  and  the  water  will 
flow  through  them  till  the  piston  reaches  its  lowest  point.  The 
same  operation  being  repeated  a  few  times,  a  column  of  water  will 
be  lifted  to  the  mouth  of  the  discharge-pipe  P,  after  which  every 
elevation  of  the  piston  will  deliver  a  volume  of  the  fluid  equal  to 
that  of  a  cylinder  whose  base  is  the  area  of  the  piston  and  whose 
altitude  is  equal  to  its  play. 

As  the  water  on  the  same  level  within  and  without  the  pump 
will  bev  in  equilibrio,  it  is  plain  that  the  resistance  to  be  overcome 
by  the  power  will  b«   the  friction  of  the  rubbing  surfaces  of  the  pump 


496 


ELEMENTS    0*     ANALYTICAL    MECHANICS 


augmented  by  the  weight  of  a  column  of  fluid  whose  base  is  the  area 
of  the  piston,  and  altitude  the  difference  of  level  between  the  surface 
of  the  water  in  the  reservoir  and  the  discharge-pipe.  Hence  the 
quantity  of  work  is  estimated  by  the  same  rule,  Equation  (738).  If 
we  omit  for  a  moment  the  consideration  of  friction,  and  take  but  a 
single  elevation  of  the  piston  after  the  water  has  reached  the  dis- 
chargtvpipe,  n  will  equal  one,  <p  will  be  zero,  and  that  equation  re- 
duces to 

Fp  =  62,5  Bp  X  b ; 

but  62,5  X  Bp  is  the  quantity  of  fluid  discharged  at  each  double 
stroke  of  the  piston,  and  b  being  the  elevation  of  the  discharge-pipe 
above  the  water  in  the  reservoir,  we  see  that  the  work  will  be  the 
same  as  though  that  amount  of  fluid  had  actually  been  lifted  through 
this  vertical  height,  which,  indeed,  is  the  useful  effect  of  the  pump 
■ioT  every   double  stroke. 

§  394. — The  Forcing-Pump 
is  a  further  modification  of 
the  simple  sucking-pump.  The 
Darrel  B  and  sleeping- valve 
E  are  placed  upon  the  top 
of  the  sucking-pipe  M.  The 
piston  F  is  without  per- 
foration and  valve,  and  the 
water,  after  being  forced  into 
the  barrel  by  the  atmospheric 
pressure  without,  as  in  the  suck- 
ing-pump, is  driven  by  the  de- 
pression of  the  piston  through 
a  lateral  pipe  H  into  an  air-, 
vessel  iVr,  at  the  bottom  of 
which  is  a  second  sleeping-valve 
E',  opening,  like  the  first,  up. 
ward.  Through  the  top  of  the 
air-vessel  a  discharge-pipe  K 
passes    air-tight,   nearly   to    the 


APPLICATIONS  497 

bottom.  The  water,  when  forced  into  the  air-veasel  by  the  de- 
scent of  the  piston,  rises  above  the  lower  end  of  this  pipe, 
confines  and  compresses  the  air,  which,  reacting  by  its  elas- 
ticity, forces  the  water  up  the  pipe,  while  the  valve  E'  is  closed  by 
its  own  weight  and  the  pressure  from  above,  as  soon  as  the. piston 
reaches  the  lower  limit  of  its  play.  A  few  strokes  of  the  piston  will, 
in  general,  be  sufficient  to  raise  water  in  the  pipe  K  to  any  desired 
height,  the  only  limit  being  that  determined  by  the  power  at  com- 
mand and  the  strength  of  the  pump. 

§  395. — During  the  ascent  of  the  piston,  the  valve  E  is  closed 
and  E  is  open  ;  the  pressure  upon  the  upper  surface  of  the  piston 
is  that  exerted  by  the  entire  atmosphere ;  the  pressure  upon  the 
lower  surface  is  that  of  the  entire  atmosphere  transmitted  from  the 
surface  of  the  reservoir  through  the  fluid  up  the  pump,  diminished 
by  the  weight  of  the  column  of  water  whose  base  is  the  area  of 
the  piston  and  altitude  the  height  of  the  piston  above  the  surface 
of  the  water  in  the  reservoir  ;  hence,  the  resistance  to  be  overcome 
by  the  power  will  be  the  difference  of  these  pressures,  which  is 
obviously  the  weight  of  this  column  of  water.  Denote  the  area 
of  the  piston  by  2?,  its  height  above  the  water  of  the  reservoir  at. 
one  instant  by  y,  and  the  weight  of  a  unit  of  volume  of  the  fluid 
by  w,  then  will  the  resistance  to  be  overcome  at  this  point  of  th*». 
ascent  be 

w  .  B .  y  ; 

and    the   elementary  quantity  of  work   will   be 

w  .  B  .ydy  ; 

and   the  whole  work   during    the   ascent  will   be 


w 


-  Bfy'ydy  =  » •*- v-^  b'  -  rJ ; 


in  which  y'    and   yt  are    the  distances  of  the  upper  and  lower  limits 
of  the   play  of  the    piston  from    the  water   in    the   reservoir. 

But  B .  (y'  —  yt)  is  the  volume  of  the  barrel  within  the  limits 
of  the  play  of  the  piston,  and  \  (y'  -f-  y,)  is  the  height  of  its  centre 
of  gravity  above   the   level  of  the  fluid    in    the    reservoir 


498  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

y'  -\-  y 
Denoting  the    play  by  p,  and   maKing    — - — -  =  z\  we   have  foi 

lit 

the  quantity  of  work    during  the   ascent, 

w  .  B  .p  .  z'. 

During  the  descent  of  the  piston,  the  valve  E  is  closed,  and  W 
open,  and  as  the  columns  of  the  fluid  in  the  barrel  and  discharge- 
pipe,  below  the  horizontal  plane  of  the  lower  surface  of  the  pistoi^ 
will  maintain  each  other  in  equilibrio,»  the  resistance  to  be  over- 
come by  the  power  will  be  the  weight  of  a  column  of  fluid  whose 
base  is  the  area  of  the  piston  and  altitude  the  difference  of  level 
between  the  piston  and  point  of  delivery  P ;  and  denoting  by  z. 
the  distance  of  the  central  point  of  the  play  below  the  point  P> 
we   shall   find,  by  exactly  the   same   process, 

w  Bp  zt , 

for  the  quantity  of  work  of  the  motor  during  the  descent  of  "he 
piston ;  and  hence  the  quantity  of  work  during  an  entire  double 
stroke  will  be   the  sum  of  these,  or 

w  Bp  (z'  4-  zt). 

But  z'  -f-  zt  is  the  height  of  the  point  of  delivery  P  above  the 
surface  of  the  water  in  the  reservoir ;  denoting  this,  as  before,  by 
6,  we   have 

w Bpb  ; 

and  calling  the  number  of  double  strokes  n,  and  the  whole  quantity 
of  work   Q,  we   finally    have 

Q  =  nw  Bpb. (730) 

b 

If  we    make    zt  =  z\    or   6  =  2^,  which  will  give   zt  ==  — »    the 

quantity  of  work  during  the  ascent  will  be  equal  to  that  during 
the  descent,  and  thus,  in  the  forcing-pump,  the  work  may  be  equalized 
and  the  motion  made  in  some  degree  regular.  In  the  lifting  and 
sucking-pumps  the  motor  has,  during  the  ascent  of  the  piston,  to 
overcome  the  weight  of  the  entire  column  whose  base  is  equal  to 
the  area  of  the   piston    and    altitude   the   difference  of  level   between 


APPLICATIONS. 


4913 


the  water  in  the  reservoir  and  point  of  delivery,  and  being  wholly 
relieved  during  the  descent,  when  tlje  l>ad  is  thrown  upon  the 
sleeping-valve  and  its  box,  the  work  becomes  variable,  and  the 
motion    irregular. 


THE    SIPHON. 


g  396. — The  Siphon  is  a  bent  tube  of  unequal  branches,  open  ai 
both  ends,  and  is  used  to  convey  a  liquid 
from  a  higher  to  a  lower  level,  over  an  in- 
termediate point  higher  than  either.  Its 
parallel  branches  being  in  a  vertical  plane 
and  plunged  into  two  liquids  whose  upper 
surfaces  are  at  L  M  and  L'  J\f,  the  fluid 
will  stand  at  the  same  level  both  within 
and  without  each  branch  of  the  tube  when 
a  vent  or  small  opening  is  made  at  0. 
If  the  air  be  withdrawn  from  the  siphon 
through  this  vent,  the  water  will  rise  in  the 

branches  by  the  atmospheric  pressure  without,  and  when  the  two 
columns  unite  and  the  vent  is  closed,  the  liquid  will  flow  from  the 
reservoir  A  to  A\  as  long  as  the  level  L'  M'  is  below  L  M,  and  the 
end  of  the  shorter  branch  of  the  siphon  is  below  the  surface  of  the 
liquid    in    the  reservoir    A. 

The  atmospheric  pressures  upon  the  surfaces  L  M  and  U  M\ 
tend  to  force  the  liquid  up  the  two  branches  of  the  tube.  When 
the  siphon  is  filled  with  the  liquid,  each  of  these  pressures  is  coun- 
teracted in  part  by  the  pressure  of  the  fluid  column  in  the  branch 
of  the  siphon  that  dips  into  the  fluid  upon  which  the  pressure  is 
exerted.  The  atmospheric  pressures  are  very  nearly  the  same  for  a 
difference  of  level  of  several  feet,  by  reason  of  the  slight  density 
of  air.  The  pressures  of  the  suspended  columns  of  water  will,  for  the 
same  difference  of  level,  differ  considerably,  in  consequence  of  the 
greater  density  of  the  liquid.  The  atmospheric  pressure  opposed 
to  the  weight  of  the  longer  column  will  therefore  be  more  counter- 
acted   than    that  opposed    to   the   weight  of   the  shorter,  thus  leaving 


500         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

an  excess  of  pressure  at  the  end  of  the  shorter  branch,  which  will 
produce  the  motion.  Thus,  denote  by  A  the  intensity  of  the  at- 
mospheric pressure  upon  a  surface  a  equal  to  that  of  a  cross-section 
of  the  tube ;  by  h  the  difference  of  level  between  the  surface  L  M 
and  the  bend  0 ;  by  A/  the  difference  of  level  between  the  same 
point  0  and  the  level  L'  M' \  by  D  the  density  of  the  liquid; 
and  by  g  the  force  of  gravity:  then  will  the  pressure,  which  tends 
to   force   the   fluid  up   the   branch   which  dips   below   L  M,  be 

A  —  ah  D g\ 

and  that  which  tends  to  force  the  fluid  up  the  branch  immersed 
in   the   other   reservoir,  be 

A  —  ah' D g  ; 

and  subtracting  the  first  from  the  second,  we  find 

aDg(h'  -  h), 

for  the  intensity  of  the  force  which  urges  the  fluid  within  the 
siphon,  from  the  upper  to   the   lower   reservoir. 

Denote  by  I  the  length  of  the  siphon  from  one  level  to  the 
other.  This  will  be  the  distance  over  which  the  above  force  will 
be  instantly  transmitted,  aiid  the  quantity  of  its  work  will  be 
measured   by 

aDg(h'  —  h)l. 

■ 

The  mass  moved  will  be  the  fluid  in  the  siphon  which  is  measured 
by  alD\  and  if  we  denote  the  velocity  by  V,  we  shall  have,  for  the 
living  force  of  the  moving  mass, 

alD.  F2; 
whence, 

aDg{h'  —  h)lz= ; 

and, 

V  =  -/20(A'-A); 

from  which  it  appears,  that  the  velocity  with  which  the  liquid  will 
flow  through  the  siphon,  is  equal  to  the  square  root  of  twice  the  force 
of  gravity,  into    the   difference   of  level   of   the  fluid  in   the    two   reser- 


APPLICATIONS.  501 

voirs.  When  the  fluid  in  the  reservoirs  comes  to  the  same  level, 
the   flow   will   cease,  since,  in  that  case,  h'  —  h  =  0.  • 

§  397. — The  siphon  may  be  employed  to  great  advantage  to 
drain  canals,  ponds,  marshes,  and  the  like.  For  this  purpose,  it  may 
be  made  flexible  by  constructing  it 
of  leather,  well  saturated  with 
grease,  like  the  common  hose,  and 
furnished  with  internal  hoops  to 
prevent  its  collapsing  by  the  pres- 
sure of  the  external  air.  It  is 
thrown  into  the  water  to  be  drained, 
and  filled ;  when,  the  ends  being 
plugged  up,  it   is   placed  across  the 

ridge  or  bank  over  which  the  water  is  to  be  conveyed ;  the  plugs 
are  then  removed,  the  flow  will  take  place,  and  thus  the  atmos- 
phere will  be  made  literally  to  press  the  water  from  one  basin  to 
another,  over   an    intermediate  ridge. 

It  is  obvious  that  the  difference  of  level  between  the  bottom  of 
the  basin  to  be  drained  and  the  highest  point  0,  over  which  the 
water  is  to  be  conveyed,  should  never  exceed  the  height  to  which 
water  may  be  supported  in  vacuo  by  the  atmospheric  pressure  at 
the   place. 

THE    AIR-PUMP. 

§  398. — Air  expands  and  tends  to  diffuse  itself  in  all  directions 
when  the  surrounding  pressure  is  lessened.  By  means  of  this  pro- 
perty, it  may  be  rarefied  and  brought  to  almost  any  degree  of  teru- 
ity.  This  is  accomplished  by  an  instrument  called  the  Air-Pump  or 
Exhausting  Syringe.  It  will  be  best  understood  by  describing  one 
of  the  simplest  kind.      It  consists,  essentially,  of 

1st.  A  Receiver  R,  or  chamber  from  which  the  exterior  air  is  ex- 
cluded, that  the  air  within  may  be  rarefied.  This  is  commonly  a 
bell-shaped  glass  vessel,  with  ground  edge,  over  which  a  small  quaj 

tity  of  grease  is  smeared,  that  no  air  may  pass  through  any  remain 

31 


502 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


fif»M^^ 


P    **»i<»^iAtt!)AMa 


ing  inequalities  on  its  surface,  and  a  ground  glass  plate  m  n  imbedded 
in  a  metallic  table,  on  which  it  stands. 

2d.  K  Barrel  B, 
or  chamber  into 
which  the  air  in 
the  reservoir  is  to 
expand  itself.  It 
is  a  hollow  cylin- 
der of  metal  or 
glass,  connected 
with  the  receiver 
R  by  the  commu- 
nication ofg.     An 

air-tight  piston  P  is  made   to  move  back   and  forth  in   the  barrel    by 
means  of  the  handle  a. 

3d.  A  Stop-cock  h,  by  means  of  which  the  communication  between 
the  barrel  and  receiver  is  established  or  cut  off  at  pleasure.  This 
cock  is  a  conical  piece  of  metal  fitting  air-tight  into  an  aperture 
just  at  the  lower  end  of  the  barrel,  and  is  pierced  in  two  directions ; 
one  of  the  perforations  runs  transversely  through,  as  shown  in  the 
first  figure,  and  when  in  this  position  the  communication  between 
the  barrel  and  re- 
ceiver is  estab- 
lished ;   the  second 


MMtesfo  ■-*  y\\\'\\»\^m%%^^to^- 


perforation  '  passes 

in  the  direction  of 

the  axis  from    the 

smaller    end,    and 

as     it     approaches 

the   first,  inclines   sideways,  and  runs   out   at   right    angles   to   it,   as 

indicated   in   the    second   figure.      In   this   position   of   the   cock,    the 

communication    between    the   receiver   and   barrel   is    cut    off,    whilst 

that  with  the  external  air  is  opened. 

Now,  suppose  the  piston  at  the  bottom  of  the  barrel,  and  the 
communication  between  the  barrel  and  the  receiver  established; 
draw  the  piston  back,  the  air    in    the   receiver   will    rush   out   in   the 


APPLICATIONS.  503 

direction  indicated  by  the  arrow-head,  through  the  communication 
ofg,  into  the  vacant  space  within  the  barrel.  The  air  which  now 
occupies  both  the  barrel  and  receiver  is  less  dense  than  when  it  occu- 
pied the  receiver  alone.  Turn  the  cock  a  quarter  round,  the  com- 
munication between  the  receiver  and  barrel  is  cut  off,  and  that  be- 
tween the  latter  and  the  open  air  is  established ;  push  the  piston  to 
the  bottom  of  the  barrel  again,  the  air  *within  the  barrel  will  be 
delivered,  into  the  external  air.  Turn  the  cock  a  quarter  back,  the 
communication  between  the  barrel  and  receiver  is  restored;  and 
:he  same  operation  as  before  being  repeated,  a  certain  quantity  of 
air  will  be  transferred  from  the  receiver  to  the  exterior  space  at 
each  double  stroke  of  the  piston. 

To  find  the  degree  of  exhaustion  after  any  number  of  double 
strokes  of  the  piston,  denote  by  D  the  density  of  the  air  in  the  re- 
ceiver before  the  operation  begins,  being  the  same  as  that  of  the 
external  air ;  by  r  the  capacity  of  the  receiver,  by  b  that  of  the  bar- 
rel, and  by  p  that  of  the  pipe.  At  the  beginning  of  the  operation, 
the  piston  is  at  the  bottom  of  the  barrel,  and  the  internal  air  occu 
pies  the  receiver  and  pipe;  when  the  piston  is  withdrawn  to  the 
opposite  end  of  the  barrel,  this  same  air  expands  and  occupies  the 
receiver,  pipe,  and  barrel ;  and  as  the  density  of  the  same  body  is 
inversely  proportional  to  the  space  it  occupies,  we  shall  have 

r  -\-  p  +  b     :     r  -+-  p     ::     D     :     x\ 

in  which  x  denotes  the  density  of  the  air  after  the  piston  is  drawn 
back  the  first  time.      From  this  proportion,  we  find 

, '"„  J) .      r+P    . 

r  -\-  p  -\-  b 

The  cock  being  turned  a  quarter  round,  the  piston  pushed  back  to 
the  bottom  of  the  barrel,  and  the  cock  again  turned  to  open  the 
communication  with  the  receiver,  the  operation  is  repeated  upon  the 
air  whose  density  is  x,  and  we  have 

r+p  +  b     :     r+p     :  :     />  .  — 1±£^     :     *'; 

in  which  x'  is  the  density  after  the  second  backward  motion  -of  fk« 
piston,  or  after  the  second  double  stroke ;    and  we  find 


504 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


x 


-»-c 


r  4-  p 


*■  +  p  +  fc 

and   if  n  denote  the  number   of  double   strokes   of   the   piston,  and 
ga  the  corresponding  density  of  the  remaining  air,  then  will 

V 


x. 


V  4-   »  +  fr/ 


+  />  +  6> 

From  which  it  is  obvious,  that  although  the  density  of  the  air  will 
become  less  and  less  at  every  double  stroke,  yet  it  can  never  be 
reduced  to  nothing,  however  great  n  may  be ;  in  other  words,  the 
air  cannot  be  wholly  removed  from  the  receiver  by  the  air-pump. 
The  exhaustion  will  go  on  rapidly  in  proportion  as  the  barrel  is 
large  as  compared  with  the  receiver  and  pipe,  and  after  a  few  double 
strokes,  the  rarefaction  will  be  sufficient  for  all  practical  purposes. 
Suppose,  for  example,  the  receiver  to  contain  19  units  of  volume,  the 
pipe  1,  and  the  barrel  10;  then  will 

r  +  p  20 


—    2  . 

30  ""  3  * 


T  -f-  p  +  b 
and    suppose  4   double    strokes   of  the  piston;    then  will  n  —  4,  and 

t^y  m  W =  Jf.=..Mw.  ***** 

that  is,  after  4  double  strokes,  the  density  of  the  remaining  air  will 
be  but  about  two  tenths  of  the  original  density.  With  the  best 
machines,  the   air   may    be  rarefied  from    four  to    six  hundred  times. 

The   degree  of  rarefaction    is   indicated   in   a    very 
simple    manner    by    what    are    called   gauges.      These  (&JR 

not  only  indicate  the  condition  of  the  air  in  the 
receiver,  but  also  warn  the  operator  of  any  leakage 
that  may  take  place  either  at  the  edge  of  the  receiver 
or  ill  the  joints  of  the  instrument.  The  mode  in 
which  the  gauge  acts,  will  be  readily  understood  from 
the  discussion  of  the  barometer;  it  will  be  suffi- 
cient here  simply  to  indicate  its  construction.  In  its 
more  perfect  form,  it  consists  of  a  glass  tube,  about  60  inches  long, 
bent  in  the  middle  till  the  straight  portions  are  parallel  to  each 
other;  one  end  is  closed,  and    the   branch  terminating  in  this    ew\  is 


£S^ 


APPLICATIONS. 


505 


filled  with  mercury.  A  scale  of  equal  parts  is  placed  between  the 
branches,  having  its  zero  at  a  point  midway  from  the  top  to  the 
bottom,  the  numbers  of  the  scale  increasing  in  both  directions.  It 
is  placed  so  that  the  branches  of  the  tube  shall  be  vertical,  with 
its  ends  upward,  and  inclosed  in  an  inverted  glass  vessel,  .  which 
communicales  with  the  receiver  of  the  air-pump. 

Repeated  attempts  have  been  made  to  bring  the  air  pump  to 
still  higher  degrees  of  perfection  since  its  first  invention.  Self-acting 
valves,  opening  and  shutting  by  the  elastic  force  of  the  air,  have 
been  used  instead*  of  cocks.  Two  barrels  have  been  employed  in- 
stead  of  one,  so  that  an  uninterrupted  and  more  rapid  rarefaction 
of  the  air  is  brought  about,  the  piston  in  one  barrel  being  made 
to  ascend  while  that  of  the   other  descends.     The  most  serious  defect 


if 


LB=£Q 


tras  that  by  which  a  portion  of  the  air  was  retained  between  the 
piston  and  the  bottom  of  the  barrel.  This  intervening  space  is  filled 
with   air    of    the   ordinary    density    at    each   descent   of    the   piston ; 


506  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

when  the  cock  is  turned,  and  the  communication  re-established  with 
the  receiver,  this  air  forces  its  way  in  and  diminishes  the  rarefac- 
tion already  attained.  If  the  air  in  the  receiver  is  so  far  rarefied, 
that  one  stroke  of  the  piston  will  only  raise  such  a  quantity  as 
equals  the  air  contained  in  this  space,  it  is  plain  that  no  further 
exhaustion  can  be  effected  by  continuing  to  pump.  This  limit  to 
rarefaction  will  be  arrived  at  the  sooner,  in  proportion  as  the 
space  below  the  piston  is  larger;  and  one  chief  point  in  the  im- 
provements has  been  to  diminish  this  space  as  much  as  possible. 
A  B  is  a  highly  polished  cylinder  of  glass,  which  serves  as  the  bar- 
rel of  the  pump  ;  within  it  the  piston  works  perfectly  air-tight.  The 
piston  consists  of  washers  of  leather  soaked  in  oil,  or  of  cork 
covered  with  a  leather  cap,  and  tied  together  about  the  lower  end 
C  of  the  piston-rod  by  means  of  two  parallel  metal  plates.  The 
piston-rod  (76,  which  is  toothed,  is  elevated  and  depressed  by  means* 
of  a  cog-wheel  turned  by  the  handle  M.  If  a  thin  film  of  oil  be 
poured  upon  the  upper  surface  of  the  piston  the  friction  will  be 
lessened,  and  the  whole  will  be  rendered  more  air-tight.  To  diminish 
to  the  utmost  the  space  between  the  bottom  of  the  barrel  and  the 
piston-rod,  the  form  of  a  truncated  cone  is  given  to  the  latter,  so 
that  its  extremity  may  be  brought  as  nearly  as  possible  into  abso- 
lute contact  with  the  cock  E\  this  space  is  therefore  rendered  indefi 
nitely  small,  the  oo?ing  of  the  oil  down  the  barrel  contributing  still 
further  to  lessen  it.  The  exchange-cock  E  has  the  double  bore 
already  described,  and  is  turned  by  a  short  lever,  to  which  motion 
is  communicated  by  a  -rod  c  d.  The  communication  G  H  is  carried 
to  the  two  plates  /  and  K.  on  one  or  both  of  which  receivers  ma) 
be  placed  ;  the  two  cocks  N  and  0  below  these  plates,  serve  to  cut 
off  the  rarefied  air  within  the  receivers  when  it  is  desired  to  leave 
them  for  any  length  of  time.  The  cock  0  is  also  an  exchange-cock, 
so  as  to  admit  the  external  air  into  the  receivers. 

Pumps  thus  constructed  have  advantages  over  such  as  work 
with  valves,  in  that  they  last  longer,  exhaust  better,  and  may  be 
employed  as  condensers  when  suitable  receivers  are  provided,  by 
merely  reversing  the  operations  of  the  exchange  valve  during  the 
motion  of  the  piston. 


TABLES. 


TABLE   I. 

THE  TENACITIES  OF   DIFFERENT  SUBSTANCES,  AND  THE  RESISTANCES 
WHICH  THEY  OPPOSE  To  DIRECT  COMPRESSION.— See  §269. 


SUBSTANCES    EXPERIMENTED    ON. 


Wrought-iron,  in  wire  from   l-20t!i  i 
to  1--301  li  of  au  inch  in  diame-  > 

tor ) 

in  wire,  l-10th  of  iin  inch  •     •     • 
in  bars,  Russian  (mean)  • 
English  (mean)    • 

hammered 

rolled  in  >lieets,  and  cut  length-  » 

wise J 

ditto,  cut  crosswise    •     •     • 
in  chains,  oval  links  6  in.  clear,  * 

iron  Is  in.  diameter  •    • 
ditto,  Bruntou's,  with  stay  across  I 

link j 

Cast  lion,  quality  No.  1 

2  .     .     .     • 

3  •     •     •     • 
Steel,  cast 

cast  and  tilted 

blistered  and  hammered  • 

shear  

raw 

Damascus • 

ditto,  once  refined   .... 

ditto,  twice  refined  .... 

Copper,  cast 

hammered 

slicct 

wire 

Platinum  wire     ........ 

Silver,  cast 

wnc 

Gold,  cast  •     •••••••• 

wire 

Brass,  yellow  (tine) 

Gun  metal  (hard) 

Tin,  ca>t    

wire         •     •     • 

Lead,  cast       •  

miiled  sheet ' 

wire 


c 
-      2 

X    fl    s 

C   =  3 
>J5  9 


6o  to  91 

36  to  43 
27 

3o 
14 

id 

2U 


6 
6 
6 


25 

to  7J 
to  8 
to  qJ 

44 
60 
5oi 

57  ' 
5o 

3i 

36 
44 
Sh 
i5 
21 
27J- 
17 

id 

9 
a 

8 
16 

2 

3 
4-5ths 

14 

I|I 


w  ■: 

c  s 
S.= 


Lame 

Telford 
Lame 

Brunei 

Mitis 

Bro  vn 

Barlow 
Hodgkinson 


Mitis 
Rennie 


Mitis 

Rennie 
Kingston 
Guyton 

Rennie 


Tredgold 
Guyton 


■ 

C'     . 

g     — t 

£  ? 

fcl ■  s"- 

■g      6" 
>=  sou 

=     -     ■- 

t&-  2. 


*  2 


«<  a. 


38  to  41 
37  to  48 
5i   to  65 


52 

46 


73 
7 
3* 


Hodgkinson 


Rennie 


•The  stronger  quality  01  cast  iron,  isji  Scotch  iron  known  as  the  Devon  Hot  Blast,  No.  3:  its  tenaci- 
ty Is  9J  tons  per  square  inch,  and  its  resistance  to  compression  65  tons.  The  experiments  of  Major 
Wade  on  the  gun  Iron  at  VVesi  Point  Foundry,  and  at  Boston,  give  results  as  high  as  10  to  16  tons,  and 
on  small  cast  bars,  as  high  as  17  tuns. — See  Ordnance  Manual,  1850,  p.  402 


>AULE    I. 


509 


TABLE  I — continued. 


SUBSTANCES     EXPERIMENTED    ON. 


■) 


Stone,  shite  (Welsh)      • 

Marble  (white)  •     • 

Givry       .... 

Portland  .... 

Craigleith  freestone 

Bra m ley  Fall  sandston 

Cornish  granite 

Peterhead  ditto 

Limestone  (compact  blk) 

Pui  beck  .... 

Aberdeen  granite    • 
Brick,  pale  red    .     •     • 

red 

Hammersmith  (pavior 
ditto      (burnt)  • 
Chilk  •     •     •     « 
[Plaster  of  Paris  •     •     • 
61ftg<&   nlute     .... 
Bone  (ox)       .... 
Hemp  fibres  glued    aether 
Strips  of  paper  gluea  together 
Wood,  Box,  spec,  gravity 

Ash 

Teak 

Beech       .... 

Oak 

Ditto 

Fir 

Pear 

Mahogany     •     •     • 

Elm     ...     •     • 

Pine,  American 

Deal,  white   •     •     • 


,862 

,6 

,9 

,7 

,92 

f 

,646 
,637 


a   -8 

C    r.    — 

CCS 


5,7 

4 

1 


,13 


,o3 
4 

2,2 
41 

i3 


5 

4 

5 

4* 

3* 

6 

6 

6 


X 

W 


n 


c  - 

■! 

z  8. 


Barlow 


■ 

c     . 
SI   -  — • 

£  *  - 

o£2L 


M 

1,6 

2,4 

2,8 

3,7 
4 
4 
5 

,56 

,8 
I 

M 
,22 


«,7 


'5? 
.7* 


e 


W  v.' 

c  s 


£* 


Kcnnie 


510 


TABLE    II. 


TABLE  H. 


OF  THE  DENSITIES  AND  VOLUMES  OF  WATER  AT  DIFFERENT  DEGREES 
OF  HEAT,  (ACCORDING  TO  STAMPFER),  FOR  EVERY  2*  DEGREES  Of 
FAHRENHEIT'S  SCALE.— See  § 276. 

(Jahrbiu     des  Polytechnischen  Institutes  in   (Vein,  Bd.  16,  S.  70). 


t 
Temperature. 

Density. 

Din". 

V 

Volume. 

Diff. 

o 

32. oo 

0,999887 

< 

i,oooii3 

34,25 

0,999950 

63 

1  ,oooo5o 

63 

36,5o 

0,999988 

38 

1,000012 

38 

38,75 

1 ,000000 

12 

1 ,000000 

12 

4i,oo 

0,999988 

12 

1,000012 

12 

43,25 

0,999952 
0,999894 

35 

1 ,000047 

35 

45, 5o 

58 

1,000106 

£ 

47,75 

0,999813 

81 

1,000187 

81 

5o.oo 

0,999711 

102 

1.000289 

102 

52,25 

0,999587 

124 

1. ooo4i3 

124 

54, 5o 

0,999442 

i45 

i,ooo558 

145 

56,75 

0,999278 

164 

1,000723 

1 65 

59,00 

0,999095 

1 83 

1 ,000906 

1 83 

61, 25 

0,998893 

202 

1,001108 

202 

63.5o 

0,998673 

220 

1, 001329 

221 

65,75 

0,998435 

238 

1, 00 1 567 

238 

68,00 

0,998180 

255 

•  1,001822 

255 

70,25 

0,997909 

271 

1,002095 

.   273 

72,5o 

0.997622 

287 

1,002384 

289 

74,75 

0,997320 

3o2 

1,002687 

3o3 

77,00 

0,997003 

3i7 

1  .oo3oo5 

3i8 

79,25 

0,996673 

33o 

i,oo3338 

333 

81, 5o 

0,996329 

344 

i,oo3685 

347 

83,75 

0,993971 

358 

1 ,004045 

36o 

86,00 

0,995601 

370 

1,004418 

373 

88,25 

0,995219 

382 

1 ,004804 

386 

90,50 

0,994825 

394 

I,O0D2O2 

398 

92,75 

0,994420 

4o5 

i,oo56i2 

410 

95,00 

0,994004 

416 

i,oo6o32 

420 

97,25 

0,993575 

425 

1,006462 

43o 

99, 5o 

1 

0,993143 

434 

1,006902 

440 

With  this  table  it  is  easy  to  find  the  specific  gravity  by  means  of  water  at  any  temperature 
Suppose,  for  example,  the  specific  gravity  S'  in  Equation  (456),  had  been  found  at  the  tempera- 
lure  of  59°,  then  would  D,i  in  that  equation  be  0,999095,  and  the  specific  gravity  of  the  body 
referred  to  water  at  its   greatest  density,  would  be  given  by 


«  =  S'  X  0,999095 


TABLE    III. 


511 


TABLE  in. 


F  THE  SPECIFIC  GRAVITIES  OF  SOME  OF  THE  MOST  IMPORTANT  BODIES. 


(The  density  of  distilled   water  is  reckoned  in  this  Table  at  its  maximum   38J°  F.  =  1 ,000]. 


Name  of  the  Body, 


Specific  Gravity. 


I.    SOLID  BODIES, 

(1)  Metals. 

Antimony  (of  the  laboratory) 

Brass       .... 

Bronze  for  cannon,  according  to  Lieut.  Mttzka 

Ditto,  mean     • 

Copper,  melted 

Ditto,  hammered     • 

Ditto,  wiYe-drawn    • 

Gold,  melted    • 

Ditto,  hammered 

Iron,  wrouglit 

Ditto,  cast,  a  mean  • 

Ditto,  gray       •         • 

Ditto,  white     • 

Ditto  for  cannon,  a  mean 

Lead,  pure  melted  • 

Ditto,  flattened 

Platinum,  native 

Ditto,  melted  • 

Ditto,  hammered  and  wire-drawn 

Quicksilver,  at  32°  Fahr 

Silver,  pure  melted 

Ditto,  hammered     • 

Steel,  cast        •         • 

Ditto,  wrought 

Ditto,  much  hardened 

Ditto,  slightly 

Tin,  chemically  pure 

Ditto,  hammered 

Ditto,  Bohemian  and  Saxon 

Ditto,  English 

Zinc,  melted    • 

Ditto,  rolled    • 


(2)  Bou.din»  Stones 


Alabaster 
Basalt      •         •   . 
Dole  rite  • 
Gneiss     • 
Granite   • 
Hornblende 
Limestone,  various 
Phonolite         • 
Porphyry 
Quartz     •         • 
Sandstone,  various 
Stones  for  building 
Syenite    • 
Tract  iy  te 
Brick 


kinds 


kinds, 


a  mean 


4,2 

—    4,7 

7,6 

—    8,8 

8,4U 

—    8,974 

8,758 

7,788 
8.878 

-    8,726 

-    8.9 

8,78 

ig.238 

—  19,253 

io,36i 

—  19>° 

7.207 

-    7,7^ 

7,25i 

7,2 

7,5 

7,21 

—    7;3o 

n,33o3 

u,388 

16,0 

—  18,94 

2o,855 

21,25 

I 3, 568 

—  i3,598 

10,474 

io,5i 

—  10,622 

7,?!9 

7,840 

7,818 

7,833 

7,291 

7,299 

-    7,475 

7,3 1 J 

7,291 
6,861 

—      7,2l5 

7>«9« 

2»2 

—    3,0 

M 

-    3,i 

2,72 

-    2,93 

2,5 

—    2,9 

2,5 

—    2,66 

2,9 

-    3,i 

2,64 

—     2.72 

»,5i 

—    2,69 

2,4 

—     2,6 

2,56 

=  \f 

2,2 

1,66 

—     2,62 

=,5 

—    3. 

2,4 

-     1,6 

i,4> 

—     i,86 

512 


TABLE    III. 


TABLE  III— Continued 


Name  of  the  Body.''     ]' 

Specific  Gravity. 

I.    SOLID  BODIES. 

(3)  Woods.       ' 

-  v *  .... 

Fresh-fH'.ecl. 

Pry. 

;  AMer      •        •        < 

0,8371 

o,5ooi 

Ash         •        •        < 

o  9036 

0  6440 

Aspen     •         •         < 

*        ■ 

'  *■"             t 

0.7654 

0.4302 

Birch       •         •         < 

0,9012 

0,6274 

Box         •        •        < 

0,9822 

0,5907 

Elm         •         •         < 

0  9476 

0.5474 
o,555o 

Fir          >         •         • 

0,8941 

Hornbeam        •         « 

0,9452 

0,7695 

,  Horse-chestnut         < 

0,8614 

0,5749 

:  Larch      •         •         < 

0,0206 

0,4735 

Lime       •         •        « 

0,8170 

0,4390 

Maple      •         •         ■ 

0.9036 

0.6592 

Oak 

1,0494 

0,6777 

Ditto,  another  specimen  < 

1.0754 

0.7075 

Pine,  Pinus  Abies  Picea  < 

0,8699 

0,4716 

Ditto,  Pinvs  Sylvestris    « 

0,9121 

o,55o2 

Poplar  (Italian)         •         < 

0,7634 

0.3931 

Willow                                 < 

0,71 55 

0,5289 
0,4873 

Ditto,  white    •        •        < 

.     ' 

0,9859 

(4)  Various  Solid  Bodiis. 

'  Charcoal,  of  cork 

0,1 

Ditto,  soft  wood      • 

0,28      —    o,44 

Ditto,  oak        .... 

1,573 

1 !   Coal                             •         •         « 

1.232      —      I,5l0 

Coke                                            • 

1,865 

Earth,  common        •         •         « 

1,48 

rough  sand         •         •         • 

1,92 

rough  earth,  with  gravel    • 

2,02 

moist  sand 

•     . 

2,o5 

gravelly  soil 

2,07 

2.l5 

clay  ..... 

• 

clay  or  loam,  with  gravel   < 

2,48 

Flint,  dark       •         • 

2,542 

Ditto,  white     .... 

2,741 

>  Gunpowder,  loosely  filled  in 

coarse  powder  »         •         < 

0,886 

nmsket  ditto     •         •         < 

0.992 

'  Ditto,  sliffhtly  shaken  down 

musket-powder 

1,069 

Ditto,  solid      .... 

2,248    —    2,563 

Ice  .... 

►                 •                 < 

0,916    —    0,9268 

Lime,  unslacked 

»                 • 

1.842 

Resin,  common 

>                 • 

1,089 

Rock-salt         •         < 

ft                 • 

2.257 

Saltpetre,  melted 

>                 • 

2,743 

Ditto,  crystallized 

i                 • 

1,900 

Slate-pencil 

i                 • 

1,8        —    2,24 

Sulphur  « 

• 

1.92      —    1,99 

Tallow     • 

I                 • 

0,942 

Turpentine 

•                 • 

0.991 

Wax,  white     • 

l                 • 

0.969 

Ditto,  yellow   ♦ 

• 

o.o65 

Ditto,  shoemaker's 

*••••••• 

0,897 

TABLE    III. 


513 


TABLE  III— Continued. 


Name  of  the  Body. 

Specific  Gravity. 

II.  LIQUIDS. 

Acid,  acetic 

l,o63 

Ditto,  muriatic 

1,211 

Diito,  nitric,  concentrated 

1,521      —     1,522 

Ditto,  sulphuric,  Enirlish 

1.845 

Ditto,  concentrated  (Nordh.) 

I,86o 

Alcohol,  free  from  water  • 

• . 

O.792 

Ditto,  common 

O.824     —     0,79 

Ammoniac,  liquid 

0,875 

Aquafortis,  double 

i,3oo 

Ditto,  single     • 

1,200 

Beer         •         • 

1  023    —    i,o3i 

Eiher,  acetic    • 

0.866 

Ditto,  muriatic 

0,845    —    0,674 

Ditto,  nitric     •         < 

0,886 

Ditto,  sulphuric 

0,715 

Oil,  linseed 

, 

>         0.928    —    0,953 

Ditto,  olive      •         « 

0.915 

Ditto,  turpentine     ■ 

0,792     —    0,891 

Ditto,  whale    •         « 

1 
0.92J 

Quicksilver      •         < 

i3.568    —  13,098 

Water,  distilled        < 

1.000 

Ditto,  rain        •         « 

1 .00 1 3 

Ditto,  sea         •         « 

i,0265  —     1,028 

Wine       •         •         < 

*               .         • 

0,992    —     i,o38 

III.  GASES. 

Kriroinetei 

W;.ter  =  1. 

3D  !•.. 

,     .      .             i 

Temp.  38JO  F. 

rrtyjsJQp 

Atmospheric  air  =  .  iQ  —     • 

' 

»         • 

0,00 i3o 

1 ,0000 

Carbonic  acid  eras      .... 

»         • 

0.00198 

1,524c 

Carbonic  oxide  gas   .... 

>   -      , 

»         . 

0.00126 

0,9069 

Carbureted  hydrogen,  a  maximum   < 

. 

0,00127 

00784 

Ditto,  from  Coals      •         •         • 

:  i 

o,ooo3o 
0,0008a 

0.J000 
0.5596 

Chlorine  •"•■••« 

0,00321 

2,4700 

Hydrio.lic  gttfl  ••"«•« 

0.00577 

4.4430 

Hydrogen        •••••« 

0.0000895 

0,0688 

Hydrosiilphuric  acid  gas  •         •         < 

,       . 

o.ooi55 

I. 1912 

Muriatic  acid  gas      .... 

0  00162 

1.2474 

Nitrogen          ..... 

• 

0,00127 

0,9760 

;  Oxygen  •••••« 

0,00143 

1,1026 

Phospliureted  hydrogen  gas    •         < 

0.001 13 

0,8700 

Steam  at  212°  Fahr. 

0.00082 

0.6235 

Snlrhurous  acid  gas 

•         < 

»         . 

0,00292 

2,2470 

p"1  - 


TABLE  IV. 


TABLE  IY. 


TABLE  FOE  FINDING  ALTITUDES.-See  §  284. 


Detached  Thermometer. 

• 

tt+t> 

A 

t,  +  t' 

A 

</  +  *' 

A 

t,  +  t> 

A 

40 

4,7689067 

75 

4,7859208 

no 

4,8022936 

145 

4,8180714 

4i 

,7694021 

76 

,7853973 

in 

,8027525 

146 

,8i85i4o 

42 

,7698971 

77 

,7868733 

112 

,8032109 

147 

,8189559 

43 

,770391 1 

78 

,7873487 

u3 

,8036687 

148 

,8193973 

44 

,77o885 I 

79 

,7878236 

114 

,8041261 

149 

,8198387 

45 

,77i3785 

80 

,7882979 

n5 

,8o4583o 

i5o 

.8202794 

46 

,7718711 

81 

,7887719 

116 

,8000393 

i5i 

,8207196 

47 

,7723633 

82 

,7892451 

U7 

,8o54953 

l52 

,b2II394 

48 

,7728548 

83 

,7897180 

118 

,8039309 

i53 

,8213958 
,8220377 

49 

,7733457 

84 

,7901903 

119 

,8o64o58 

i54 

5o 

,7738363 

85 

,7906621 

120 

,8068604 

1 55 

,8224761 

5i 

,7743261 

86 

,7911335 

121 

,8073144 

•  106 

,8229141 

5a 

,7748i53 

87 

,7916042 

122 

,8077680 

1 57 

.8233317 

53 

,7753o42 

88 

,7920745 

123 

,8082211 

1 58 

,6237888 

54 

,7757925 

89 

,7925441 

124 

,8086737 

139 

.6242236 

55 

,7762802 

90 

,793oi  35 

125 

,8091238 

160 

,8246618 

56 

,7767674 

91 

,7934822 

126 

,8095776 

161 

,8230976 

57 

,7772540 

92 

,7939504 

127 

,8100287 

162 

,b25533i 

58 

,7777400 

93 

,7944182 

128 

,8104795 

i63 

,6239680 

59 

,7782256 

94 

,7948854 

129 

,8109298 

164 

,6264024 

6o 

,7787105 

95 

,7953521 

i3o 

,8113796 

163 

,8268365 

6i 

,7791949 

96 

,7958184 

i3i 

,8118290 

166 

,6272701 

6a 

,7796788 

97 

,7962841 

132 

,8122778 

167 

,8277034 

63 

,7801622 

98 

,7967493 

i33 

,8127263 

168 

,82Si362 

64 

,7806430 

99 

,7972141 

i34 

,8131742 

169 

,8285685 

65 

,7811272 

100 

,7976784 

i35 

,8i362i6 

170 

.8290005 

66 

,7816090 

101 

,7981421 

i36 

,8140688 

Hi 

,8294319 

u 

,7820902 

102 

,7986034 

137 

,8i45i53 

172 

,829^-629 

,7825709 

io3 

,799068 1 

i38. 

,8149614 

173 

,8302937 

69 

,783o5u 

104 

,79953o3 

•  i39 

,8154070 

174 

,8307238 

70 

,78353o6 

io5 

,7999921 

■  140 

,81 58523 

175 

,83 11 536 

71 

,7840098 

106 

,8oo4333 

141 

,8162970 

176 

.83i583o 

Ti 

,7844883 

107 

,8009142 

142 

,8167413 

177 

,8320119 

73 

,7849664 

108 

,8013744 

143 

,8171852 

178 

,8324404 

74 

4,7854438 

109 

4,8oi8343 

144 

4,8176285 

179 

4,8328686 

TABLE  IV. 


515 


TABLE  YV— continued. 


WITH  THE  BAROMETER.— See  §  284. 


Latitude. 

Attached  Thermometer. 

¥ 

B 

T—  T' 

c 

c# 

0° 

0,0011689 

— 

+ 

3 

,0011624 

0° 

0,0000000 

0,0000000 

6 

,001 1433 

I 

,0000434 

9,9999566 

9 

,0011 1 17 

2 

,0000869 

,99991 3 1 

12 

,0010679 

3 

,oooi3o3 

,9998697 

i5 

,0010124 

4 

,0001738 

,9998263 

18 

,0009439 

5 

,0002172 

,9097829 
,9997395 

21 

,0008689 

6 

,0002607 

24 

,0007820 

7 

,ooo3o4r 

,9996961 

27 

,0006874 

8 

,0003476 

,9096527 

3o 

,0000848 

9 

,0003910 

,9996093 

33 

,0004738 

10 

,0004345 

,9995659 

36 

,ooo36i5 

11 

,0004780 

,9995225 

39 

,ooo2433 

12 

,ooo52i5 

,9994792 

42 

,0001223 

i3 

,ooo565o 

,9994358 

45 

,0000000 

14 

,0006084 

,9993924 

48 

.   9'9998775 

i5 

,0006319 

,9993490 

49 

,9998372 

16 

,ooo6q54 

,0993057 

5o 

,9997967 

17 

,0007389 

,9992623  • 

Si 

,9997366 

18 

,0007824 

,9992190 

52 

53 

,9997167 
,9996772 

19 

20  9 

,ooo82  5o 
,0008696 

,9991756 
,9991 323 

54 

,9996381 

21 

,0009 1 3c 

,9990889 

55 

,9990993 

Ti 

,0009565 

,9990456 

56 

,9995613 

23 

,00 1 0000 

,9990023 

57 

,9995237 

24. 

,ooio436 

,9989589 

58 

,9994866 

25 

,0010871 

,9989 1 56 

59 

,9994502 

26 

,001 i3o6 

,9988723 

6o 

,9994144 

27 

,00 i 1742 

,9988290 

63 

,999.3 1 1 5 

28 

,0012177 

,9987867 

66 

,9992161 

29 

,ooi26i3 

,9987424 

2 

,9991293 
,9989832 

3o 
3i 

,00 1 3o48 
0,00 1 3484 

,9986991 
9,9986558 

^9988834 

* 

90 

9,9988300 

I 

516 


TABLE    V. 


TABLE   V. 

COEFFICIENT  VALUES,  FOR  THE  DISCHARGE  OF  FLUIDS  THROUGH  THIN 
PLATES,  THE  ORIFICES  BEING  REMOTE  FROM  THE  LATERAL  FACES 
OF  THE  VESSEL.— See  §  300. 




• 

Head  of  fluid 

above  the 

centre  of  the 

orifice,  in  feet. 

Values  of  the  coefficients  for  orifices  whose  smallest  dimensions  or 

diameters  are — 

ft- 
0,66 

ft- 
o,33 

ft- 
0,16 

ft. 
0,08 

ft- 
0,07 

ft 
o,o3 

o,o5 
0,07 
o,i3 
0,20 
0,26 
o,33 
0,66 
1,00 
1,64 
3,28 
5,oo 
1           6,65 
32,75 

0,593 
0,596 
0,601 
0,602 
o,6o5 
o,6o3 
0,602 
0,600 

0,592 
0,602 
0,608 
o,6i3 
0,617 
0.617 
o,6i5 
0,612 
0,610 
0,600 

0,618 
0,620 
0.625 
o,63o 
o,63  r 
o,63o 
0,628 
0,626 
0,620 
0,61 5 
0,600 

0,627 
o,632 
0,640 
o,638 
o,63i 
o634 
o,632 
o,63o 
0,628 
0.620 
o,6i5 
0,600 

0,660 
0,657 
o.656 
o,655 
o,655 
0.604 

0.644 
0.640 
o,633 
0  621 
0,610 
0,600 

0,700 
0.696 
0.6S5 
0.677 
0.672 
0.667 
o,655 
o,65o 
0,644 
o.632 
0.618 
0,610 
0,600 

In  the  instance  of  gas,  the  generating  head  is  always  greater  than  6,63  fu,  and  the  coefficient  0,6 
«r  0,61,  is  taken  in  all  cases. 

For  orifices  larger  than  0,66  ft.,  the  coefficients  are  taken  as  for  this  dimension  ;  for  orifices  smallei 
*han  0,03  ft.,  the  coefficients  are  the  same  as  for  this  latter;  finally,  for  orifices  between  those  of  tht 
table,  we  lake  coefficients  whose  values  are  a  mean  between  the  latter,  corresponding  to  the  given  head 


TABLE   VI. 


517 


TABLE  VI. 

EXPER  MENTS  ON  FRICTION,  WITHOUT  UNGUENTS.    BY  If.  MORIN. 

The  surHtcea  of  friction  were  varied  from  o,o3336  to  2,7987  square  feet,  the  pressures  from 
88  lbs.  to  22  >5  lbs.,  and  the  velocities  from  a  scarcely  perceptible  motion  to  9,84  feet  per 
second.  Tne  surfaces  of  wood  were  planed,  and  those  of  metal  filed  and  polished  with  the 
greate>t  care,  and  carefully  wiped  after  every  experiment.  The  presence  of  unguents  was 
•specially  guarded  agaiust.— See  §  855. 


SURFACES  OF  CONTACT. 


Oak  upon  oak,  the  direction  of  the  fibres 
beiiiir  parallel  t<>  the  motion      •     • 

Oak  upon  oak,  the  directions  of  the  fibres 
of  tne  moving  surface  being  perpen- 
dioular  to  those  uf  the  quiescent  sur- 
face and  to  the  direction  of  the  motion.}: 

Oak  upon  oak,  the  fibres  of  the  both  sur- 
faces being  perpendicular  to  the  direc- 
tion of  the  motion 

Oak  upon  oak,  the  fibres  of  the  moving 
surface  being  perpendicular  t»>  the  sur- 
face of  contact,  and  those  of  the  surface 
at  rest  pantile!  to  the  direction  of  the 
motion 

Oak  upon  oak.  tliefinresof  both  surfaces 
being  perpendicular  to  the  surface  of 
contact,  or  the  pieces  end  to  end   • 

Elm  upon  oak,  the  direction  of  the  fibres 
behi<r  parallel  to  the  motion 

Oak  upon  elm,  ditto§ 

Elm  upon  oak,  the  fibres  of  the  moving 
surface  (theelin)  being  perpendicular  to 
those  of  the  quiescent  surface  (the  oak) 
and  to  the  direction  of  the  motion* 

Ash  upon  oak,  tiie  fibre*  of  both  surfaces 
beinif  parallel  to  the  direction  of  the 
motion 

Fir  upon  oak,  the  fibres  of  both  surfaces 
b-iug  parallel  to  the  direction  of  the 
motion   • 

Bead  1  upon  oak.  ditto 

Wild  pear-tree  upon  oak,  ditto     • 

Service-tree  upon  oak,  ditto    .... 

"Wrought  iron  upon  oak,  ditto|  • 


Friction  or 
Motion.* 


5  v. 


0,478 

o,324 


o,336 


0,192 


o,432 
0,246 

o.45o 


0,400 


o  355 

o,36o 
0,370 
0.400 
0,619 


a 

=  -  x 

5  £  I 


25°  33 ' 

17  58 

18  35 
10  52 


I  RICTlON    or 
ftVUMCKM  K.f 


23  2: 
i3  5c 

24  16 

21  49 

19  33 

19  48 

20  19 

21  49 
3i  47 


Jl 

&  — 


0,625 
0,540 


0,271 


0.43 

0,694 
0,376 

0,570 


0,570 


0,020 


0,53 

o  44o 

0.570 
0,619 


it 


—  -  « 


32°  1' 

28  23 


i5  10 


23  17 


34 

46 

20 

37 

29 

41 

29 

41 

27 

29 

27 
2J 

3i 

56 
45 
4( 

47 

•  The  friction  In  this  case  varies  hut  verv  slightly  from  the  mean. 

t  The  friction  in  thU  c  ise  varies  considerately  from  the  meitn.  In  all  the  exj)eriments  the  MNMf  $ 
had  lietn  LI  ninuiie>  in  mat  tct.  ,  , 

t  Th-  diunn  in. is  of  I  be  surfaces  of  contact  were  In  this  experiment  .947  square  feet,  ard  the  results 
were  nearlv  uniform.     When  thedimensions  were  iliminisheil  to  .043  a  tearing  of  tie  fit  re  Im  1 ine  appa- 
rent in  thee   -e  of  mntin:i,  and  there  were  symptoms  of  the  emwli— I IW  of  the  wood:    Ironi    IIM  cir 
eumst   nee-  there  re-nlt-d  «a  irregularity  in  the  friction  indicative  of  BSCSM Iw  pTWMWB. 

$  It  is  worthy  of  rem-. rk  that  tie  friction  of  oak  ii|M»n  elm  i*  Inn  five-i  ir.tb*  ol  ih.tof  elm  ii|H,n  oak. 

||  In  the  experiments  in  which  me  of  the  surfaces  w.is  of  Met  I.  Mini  I  particles  of  the  metal  began, 
after  a  time,  10  be  appoeni  ii|mmi  the  wood,  giving  it  a  polished  m-iallic  appaaraar*  these  were  at  every 
experimen  \\ip<dor»';  tiny  indicated  a  we  ring  of  the  metal.  The  friction  of  motion  and  that  ol  quio*- 
cence,  in  these  experiments,  coincided.    The  results  were  remarkably  unilirm. 

32 


513 


TABLE    VI. 


TABLE  YI— continued. 


SURFACES  OF  CONTACT. 


Friction  of 
Motion. 


•f 


Wrought  iron  upon  oak,  the  surfaces 
being  greased  and  well  wetted* 

Wrought  iron  upon  elm 

Wrought  iron  upon  cast  iron,  the  fibres  ) 
of  the  iron  being  parallel  to  the  motion  \ 

Wrought  iron  upon  wrought  iron,  the  ) 
fibres  of  both  surfaces  being  parallel  • 
to  the  motion ) 

Cast  iron  upon  oak,  ditto 

Ditto,  the  surfaces  being  greased  and  I 
wetted  •     • j 

Cast  iron  upon  elm  •     •     •     •.     • 

Cast  iron  upon  cast  iron 

Ditto,  water  being  interposed  between  ) 
the  surfaces ) 

Cast  iron  upon  brass 

Oak  upon  cast  iron,  the  fibres  of  the  wood  j 
being  perpendicular  to  the  direction  > 
of  the  motion ) 

Hornbeam  upon  cast  iron — fibres  paral-  i 
lei  to  motion j 

Wild  pear-tree  upon  cast  iron — fibres  j 
parallel  to  the  motion      •     •     •     •     •  \ 

Steel  upon  cast  iron 

Steel  upon  brass ■    • 

Yellow  copper  upon  cast  iron  .... 
Ditto  oak      .... 

Brass  upon  cast  iron 

Brass  upon  wrought  iron,  the  fibres  of) 
the  iron  being  parallel  to  the  motion  •  j 

Wrought  iron  upon  brass 

Brass  upon  brass 

Black  leather  (curried)  upon  oak*    • 

Ox  hide  (such  as  that  used  for  soles  and  J 
for  the  stuffing  of  pistons)  upon  oak,  V 

rou<rh j 

Ditto        ditto        ditto    smooth     • 

Leather  as  above,  polished  and  hardened  ) 
by  hammering j 

Hempen   girth,  or  pulley-band,  (sangle" 
de  chanvre,)  upon  oak,  the  fibres  of 
the  wood  and  the  direction  of  the  cord 
being  parallel  to  the  motion     • 

Hempen  matting,  woven  with  small  1 
cords,  ditto. j 

Old  cordage,  li  inch  in  diameter,  dittof 


c  z 
» -z 

fl 

Go 


0.256 

0,232 
0,194 

0,1 38 
0,490 

•  • 

0,195 

0,1 02 

0.3 1 4 
o,i47 

0,372 

0,394 

o,436 

0,202 
0,1 52 
0.189 
0,617 
0.217 

0,161 

0,172 
0,201 
0.265 

0,52 

o,335 
0.296 

0,52 

o,32 

0,52 


6t  -  = 

c  -  * 
-  ws 

.=   -   4) 


14°  22' 

14  9 

10  5g 

7  D2 

26  7 

•  • 

11  3 

8  39 

17  26 
8  22 

20  25 

21  3i 

23  34 

11  26 

8  39 

10  49 
3i  4> 

12  i5 

9  9 

9  46 

1 1  22 
14  5i 

27  29 

18  3i 

16  3o 

27  29 

17  45 
27  29 


Friction  or 
Quiescknck. 


c  s 

c  3 


0.649 

0,194 
0,137 

0,64^) 
0,162 


0,617 


0,74 

o,6o5 
o,43 

•  • 

0,64 

o,5o 
o,79 


■ 

ttTT  a 

C    "*    23 

2'S  * 

=  W.J 

.2   =   0> 


33°    o' 

10    59 

7    49 

32      52 

9    i3 


3i     41 


36  3i 

3i  11 

23  17 

•  • 

32  38 

26  34 

38  19 


•  The  friction  of  motion  was  very  nearly  the  same  whether  the  surface  of  contact  was  the  inside 
;»r  the  outside  of  the  skin. — The  constancy  of  the  coefficient  of  the  friction  of  motion  was  equally  ap- 
parent in  the  rough  and  the  smooth  skins. 

t  All  the  above  ex|ieriments,  except  that  with  curried  black  leather,  presented  the  phenomenon  of 
<»  change  in  the  polish  of  the  surfaces  of  friction — a  state  of  their  surfaces  necessary  tn,  and  dependent 
apon.  their  motion  upon  one  another. 


TABLE   VI. 


519 


TABLE  Yl—contii  ued. 


SURFACES  OF  CONTACT. 


Calcareous  oolitic  stone,  used  in  building, 
of  a  moderately  hard  quality,  called 
stone   of    Jaumont — upon    the   same 

stone 

Hard  calcareous  stone  of  Brouck,  of  a 
light  gray  color,  susceptible  of  taking 
a  tine  polish,  (the  mu>chelkalk,)  mov- 
ing upon  the  same  stone 

The  soft  stone  mentioned  above,  upon 

the  hard     

The  hard  stone  mentioned  above  upon 

the  soft 

Common  brick  upon  the  stone  of  Jaumont 

Oak  upon  ditto,  the  fibres  of  the  wood 

being  perpendicular  to  the  surface  of 

the  stone • 

Wrought  iron  upon  ditto,  ditto    • 
Common  brick  upon  the  stone  of  Brouck 
Oak  as  before  (endwise)  upou  ditto  • 
lion,  ditto  ditto      •     • 


Friction  of 
Motion. 


—  c 

c  S 

6<s 


0,64 


o,38 


o,65 

0,67 
o,65 

o,38 

0,69 
0,60 
0.38 
0,24 


=  -  S 

Z.    H>*N 
.=    =    V 


3a°  38' 


20  49 

33  2 

33  5o 

33  2 

20  49 

34  37 
3o  58 
20  49 
i3  3o 


Friction  of 

Ul'ltMlt.M  t. 


2  J 
0* 


o,74 


0,70 


0.75 

0,75 
o,65 

o,63 

0,49 
0,67 
0,64 
0,42 


■ 

B 
-1  --  s 


-  -i  /. 

~  -  o> 

•—  ^.  — 


36°  3i' 


35 


36  53 

36  53 

33  2 

32  i3 

26  7 

33  5o 
3a  38 
22  47 


520 


TABLE  VIL 


TABLE  VII. 

EXPERIMENTS  ON  THE  FRICTION  OF  UNCTUOUS  SURFACES. 

BY  M.  MORIN.— See  §£55. 

In  these  experiments  the  surfaces,  after  having  been  smeared  with  an  unguent,  \ver« 
wiped,  so  that  no  interposing  layer  of  the  unguent  prevented  their  intimate  contact. 


Friction  of 

Friction  or 

SURFACES  OF  CONTACT. 

Motion. 

Q.11KSCKNCK. 

*j  — 

u 

•«  s 

■ 

£  -5 

Z  ■- 

•5  : 

US. 

i  2* 

t^ 

=  »"3 

X  ■■ 

=  8  "t 

r*  *z 

r       ^ 

'~  *?  ae 

w    3 

w      w 

Oak  upon  oak,  the  fibres  being  parallel  t< 
the  motion        .... 

?(      0,108 

6° 

10' 

0,3gO 

21°     19' 

Ditto,  the  fibres  of  the  moving  body  be 
intr  perpendicular  to  the  motion* 

" £      °>'43 

8 

9 

0,3  14 

17      26 

Oak  upon  elm,  fibres  parallel' 

o.i  36 

7 

45 

Kun  upon  oak,  ditto 

0,119 

6 

48 

0,420 

22      47 

beech  upon  oak.  ditto  •          •          • 

o,33o 

18 

16 

Elm  upon  elm,  ditto 

0, 1 40 

7 

59 

Wrought  irou  upon  elm,  ditto         • 

o.i38 

7 

52 

Ditto  upon  wrought  iron,  ditto 

0.177 

10 

3 

Ditto  upon  cast  iron,  ditto     • 

• 

• 

0,Il8 

6     44 

Cast  iron  upon  wrought  iron,  ditto 

0.143 

8 

9 

Wrought  iron  upon  brass,  ditto     • 

0.160 

9 

6 

Brass  upon  wrought  irou 

0.166 

9 

26 

Cast  iron  upon  oak,  ditto 

0  107 

6 

7 

0,100 

5    43 

Ditto  upon  elm,  diuo,  the  unguent  beiiu 
tallow         ..... 

T  ( 

f         O.I  2D 

7 

8 

Ditto,  ditto,  the   unguent  being  hog': 

j'      °>,37 

lard  and  black  lead     • 

7 

49 

Elm  upon  cast  iron,  fibres  parallel  • 

o.i35 

7 

42 

O,093 

5    36 

Cast  iron  upon  cast  iron         •         • 

0,144 

8 

12 

Ditto  upon  brass                                 . 

0.1 3a 

7 

32 

Brass  upon  cast  iron              • 

0.107 

6 

7 

Ditto  upon  bra>s            •         • 

0.1 34 

7 

3S 

0,l64 

9     '9 

Copper  upo  i  oak             •          • 

0.100 

5 

43 

Yellow  copper  upon  cast  iron 

0.1 15 

6 

34 

Leather  (ox  hi  le)  well  tanned  upon  cast 
iron,  wetted       • 

'  '       0,229 

12 

54 

0,267 

U    57 

Ditto  upon  brass,  wetted       •         • 

0,244 

i3 

43 

TABLE    VIII. 


521 


TABLE  VIII. 

EXPERIMENTS  ON  FRICTION  WITH  UNGUENTS  INTERPOSED.    BY  M.  MORIN. 

Tlie  extent  of  the  surfaces  in  these  experiments  bore  such  a  relation  to  tlie  pressure,  a* 
to  cau-e  tlicin  to  be  separated  from  one  another  throughout  by  an  interposed  stratum  of 
the  unguent.— See  §  855. 


SURFACES  OF  CONTACT. 


Oak  upon  oak,  fibres  parallel 

Ditto        ditto 

Ditto        ditto 

Ditto,  fibres  perpendicular 

Ditto        ditto 

Ditto        ditto 

Ditto  upon  dm,  fibres  parallel 

Ditto        ditto 

Ditto        ditto 

Ditto  upon  cast  iron,  ditto 

Ditto  upon  wi ought  iron,  ditto 
Beech  upon  oak,  ditto 
Elm  upon  oak,  ditto  • 

Ditto        ditto         •         ♦ 

Ditto        ditto 

Ditto  upon  elm,  ditto     • 

Ditto  upon  cast  iron,  ditto 

Wrought  iron  upon  oak,  ditto 

Ditto  ditto  ditto  • 
Ditto  ditto  ditto  • 
Ditto  upon  elm,  ditto  • 
Ditto  ditto  ditto  ■ 
Ditto  ditto  ditto  • 
Ditto  upon  cast  iron,  ditto 
Ditto  ditto  ditto  • 
Ditto  ditto  ditto  • 
Ditto  upon  wrought  iron,  ditto 
Ditto  ditto  ditto  • 
Ditto  ditto  ditto  - 
Wrought  iron  upon  brass,  fibre 

Kiraliel  •'•"!•• 
itto        ditto        ditto  • 
Ditto        ditto        ditto  • 
Cast  iron  upon  oak,  ditto  • 

Ditto        ditto        ditto  • 

Ditto        ditto  ditto  • 

Ditto        ditto  ditto  • 

Ditto        ditto  ditto  • 

Ditto  upon  elm,  ditto  • 

Dit'o        ditto  ditto  • 

Ditt5        ditto        ditto  • 

Ditto,  ditto  upon  wrought  iron 
Cast  iron  upon  cast  iron     • 
Ditto        ditto 


Friction 

or 
Motion. 


-  ■ 


0,164 
0,073 
0,067 
o,o83 
0,072 

0.230 

o.  i36 
o  073 
0.066 
0.080 
0.098 
o,o55 
0.137 
0.070 
0.060 
0.139 
0,066 

o,256 

0.214 

o.o85 

0,078 

0.076 

o.o55 

o,i  o3 

0.076 

0,066 

0,082 

o,©8i 

0,070 

o,io3 

0.075 
0.078 

0,189 

0,218 

0,078 
0,075 
0,075 
0.077 
o  061 

0,091 


o,3 1 4 
0.197 


Friction 

OF 
QulESCKNCK. 


e  c  5 

c       fc. 


0,440 
0,164 

•  • 

0,254 


0,178 

0,411 
0,142 

0,217 

•  • 

0,649 


0,108 


0,100 

•  • 

o,n5 


•  • 


0,646 


0,100 


0,IOO 


•  • 


O,I0O 


•  • 


UNGUENTS. 


Drv  soap. 
Tallow. 
lh'g"s  lard. 
Tallow. 
Hog's  lard. 
'W  ater. 
Drv  soap. 
Tallow. 
Hog's  laid. 
Tallow. 
Tallow. 
Tallow. 
Dry  soap. 
Tallow. 
Hog's  lard. 
Drv  soap. 
Tallow. 

(Greased,  nnd 
saturated  with 
water. 
Drv  soap. 
Tallow. 
Tallow. 
Hog'.-  lard. 
Olive  oil. 
Tallow. 
Hog's  lard. 
Olive  oil. 
Tallow. 
1  loir's  lard. 
Olive  oil. 

Tallow. 

Hotr's  lard. 
Olive  oil. 
Drv  soap. 

(Greased,  and 
saturated  with 
water. 
Tallow. 
Ilo<r's  lard. 
Olive  oil. 
Tallow. 
Olive  oil. 
t  Hog's  lard  and 
I  plumbago. 
Ta.low. 
Water. 
Soap. 


522 


TABLE    VIII. 


TABLE  VIII.— continued. 


Friction 

Friction 

1 

or 

OF 

0 

SURFACES  OF  CONTACT. 

Motion 

Q.U1KSCKNCE. 

1 

UNGUENTS. 

<*-» 

krt 

1  i 

c        • 

.2?  '   S 
'u  %_  .2 

£c; 

E  *  0 

,8    £ 

V         T. 

i?      En 

■o 

O 

Cast  iron  upon  cast  iron     • 

0,IOC 

0,IOO 

Tallow. 

Ditto        ditto 

0,070 

0,100 

Hogs'  lard. 

Ditto        ditto 

0,064 

Olive  oil. 

Ditto        ditto        •         .     *   . 

o,o55 

j  Lard  and 
j  plumbago. 

Ditto  upon  brass    •         • 

0,1  o3 

Tallow. 

Ditto        ditto 

0,073 

Hogs'  lard. 

Ditto        ditto 

0,078 

•Olive  oil. 

Copper  upon  oak,  fibres  parallel 
Yellow  copper  upon  cast  iron     • 

0,069 

0,100 

Tallow. 

0,072 

0,10? 

Tallow. 

Ditto        ditto 

0,068 

Hogs'  lard. 

Ditto        ditto         • 

0.066 

Olive  oil. 

Brass  upon  cast  iron  • 

0.086 

0,106 

Tallow. 

Ditto        ditto 

0.077 

Olive  oil. 

Ditto  upon  wrought  iron 

0,081 

Tallow. 

Ditto        ditto 

0,089 

1  Lard  and 
j  plumbago. 

Ditto        ditto 

0.072 

Olive  oil. 

Ditto  upon  brass     • 

o,o58 

Olive  oil. 

Steel  upon  cast  iron   •         • 

0.  io5 

0,108 

Tallow. 

Ditto        ditto 

0.081 

Hogs'  lard. 

Ditto        ditto 

0,079 

Olive  oil. 

Ditto  upon  wrought  iron 

0,093 

Tallow. 

Ditto        ditto 

0,076 

Hogs'  lard. 
Tallow. 

Ditto  upon  brass     • 

o,o56 

Ditto        ditto 

o,o53 

Olive  oil. 

Ditto        ditto 

0,067 

(  Lard  and 
j  plumbago. 
1  Greased,  and 

Tanned  ox  hide  upon  cast  iron 

o,365 

•<  saturated  with 
f  water. 

Ditto        ditto 

0,1 59 

Tallow. 

Ditto        ditto 

0,1 33 

0,122 

Olive  oil. 

Ditto  upon  brass    • 

0,241 

Tallow. 

Ditto        ditto 

0,191 

Olive  oil. 

Ditto  upon  oak,     • 
Hempen  fibres  not  twisted,  mov-~ 

0,29 

0,7c 

Water. 

ing  upon  oak,  the  fibres  of  the 
hemp  being  placed  in  a  direc- 
tion perpendicular  to  the  direc- 

o,332 

0,869 

|  Greased,  and 
•<  saturated  with 
|  water. 

tion  of  the  motion,  and  those 

of  the  oak  parallel  to  it  • 
The  same  as  above,  moving  upon  ^ 
cast  iron        •       .  •         •         •  ) 

0,194 

•               • 

Tallow. 

Ditto     ..... 

0,1 53 

•  .       • 

Olive  oil. 

Soft  calcareous  stone  of  Jaumont" 

upon  the  same,  with  a  layer  of 

mortar,  of  sand,  and  lime  inter-  V 

•             • 

°>74 

posed,  after  from  10  to  15  min- 

utes' cortact. 

TABLE    IX 


523 


TABLE    IX. 

FRICTION  OF  TRUNNIONS  IN  THEIR    BOXES.-See  §  361. 


KINDS  OF  MATERIALS. 


Trunnions  of  cast  iron  and 
boxes  of  cast  iron. 


STATE  OF  SURFACES. 


Trunnions  of  cast  iron  and 
boxes  of  brass. 


Trunnions  of  castiron  and 
boxes  of  liguum-vitee. 


Trunnions  of  wrought  iron 
and  boxes  of  cast  iron. 


Trunnions  of  wrought  iron 
and  boxes  of  brass. 


Trunnions  of  wrought  iron  ( 
and  boxes  of  lignum-vi-  -J 
tee.  { 

Tr-i unions  of  brass  and  ( 
boxes  of  brass.  1 

Trunnions  of  brass  and  ) 
boxes  of  cast  iron.  J 

Trunnions  of  lignum-vitse  j 
and  boxes  of  cast  iron.    ) 

Trunnions  of  lignum-vitse  i 
and  boxes  of  lignum-  > 
vita.  J 


Unguents  of  olive  oil,  hogs'  lard, 
and  tallow     .... 

The  same  unguents  moistened  with 
water     ..... 

Unguent  of  asphaltum 

Unctuous ..... 

Unctuous  and  moistened  with  wa- 
ter        ..... 

Unguents  of  olive  oil,  hogs'  lard, 
and  tallow      .... 

Unctuous  ..... 
Unctuous  and  moistened  with  wa- 
ter        ..... 

Very  slightly  unctuous 

Without  unguents     • 

Unguents  of  olive  oil  and  hogs'  | 

lard        •         •         •  .  , 

Unctuous  with  oil  and  hogs'  lard 
Unctuous  with  a  niixture  of  hours' 

lard  and  plumbago 

Unguents  of  olive  oil,  tallow,  and 
hogs'  lard      .... 

Unguents  of  olive  oil,  hogs'  lard, 
and  tallow      .... 

Old  umruents  hardened     • 
Unctuous  and  moistened  with  wa- 
ter        ..... 
Very  slightly  unctuous 

Unguents  of  oil  or  hogs'  lard     • 
Unctuous  •         .         •         .         • 

Unguent  of  oil- 
Unguent  of  hogs'  lard 

Unguents  of  tallow  or  of  olive  oil 

Ungrucnts  of  hogs'  lard 
Unctuous.         .... 

Unguent  of  hogs'  lard 


Ratio  of  friction  to 
pressure  when  the 
unguent  is  renewed. 


By  the 
onlinnry 
method. 


i  r  i 

I  0,08  ) 


0,08 

o.o54 

0,14 

o.ii 
(  0,07 


I 


0,07    i 
to     \ 

0.08  ) 


0,16 
0.16 

0,10 
0,l8 


O.I9 
0,23 

0,11 

O.I9 

o.  10 
0.09 


0,12 

0,1 5 


Or.  con- 
tinuously 


o,o54 


o,o54 
0,034 

•  • 


o,o54 


•  • 


0,090 


o,o54 
o,o54 


o.o.'»5 

to 

o,o5a )  ! 
o,c7 


524 


TABLE     X. 

TABLE     X. 


OF  WEIGUTS  NECESSARY  TO  BEND  DIFFERENT  ROPES  AROUND  A  WJ  EEL 

ONE  FOOT  IN  DIAMETER.-- See  §  =357. 

No.  1.    White  Ropes — new  and  dry. 
Stiffness  proportional  to  the  square  of  the  diameter. 


Diameter  of  rope 
in  inch  s. 

Natural  stiffness, 
or  value  of  K. 

1 

Stiffness  for  load  of] 
1  lb.,  or  value  of  /. 

0.39 

0  79 
1,57 

3,i5 

lbs. 

0,4024 

1,6097 

6,4389 

25,7553 

lbs. 

0.0079877 
o,o3i95oi 
0,1278019 
o,5i 12019 

No.  2.    White  Ropes-  -new  and  moistened    with 

water. 

Stiffness  proportional  to  square  of  diameter. 


Diameter  of  rope 
in  inch.es. 

Natural  stiff  ess, 
or  value  of  K. 

Stiffness  for  load  ol 
1  lb.,  or  value  of  /. 

0,39 

0,79 

1,57 

3,i5 

lbs. 

O.8048 

3,2194 

12.8772 

5i,5iii 

lbs. 

0,0079877 
o,o3i95oi 
0.1278019 
o,5i 12019 

1 

No.  3.    White  Ropes — half  worn  and  dry. 

Stiffness  proportional  to  the  square  root  of  the  cube  of 
the  diameter. 


Diameter  of  ro|ie 
in  inches. 

Natural  Stiffness, 
or  value  of  K. 

Stiffness  for  load  of  j 
1  lb.,  or  value  of  /. 

O.39 

0,79 

1.57 

3,i5 

lbs. 

0.4O243 
i,i38oi 

3,21844 
9  ioi5o 

lbs. 

0.0079877 
0,o525889 
0,o638794 
0,l8o6573 

No.  4.    White  Ropes— half  worn   and  moistened 

WITH    WATER. 

Stiffness  proportional  to  the  square  root  of  the  cul/e  of 
the  diameter. 


Diameter  of  rope 
in  inches. 

Natural  Stiffness, 
or  value  of  K. 

Stiffness  for  load  of 
1  lb.. or  value  of  I. 

o,39 
0.79 
1, 37 
3,i5 

1 

lbs. 

0,8048 
2  2761 
6,4324 

18,2037 

lbs. 

0,0079877 

o,o525889 

0,0638794 

0,1806573 

1 

Squ  1  res  of  1  he  ratios 

ol  diameter,  or  Tal 

ties  of  (/%. 

Squ  res 

Ratios  (J. 

d*. 

1,00 

1. 00 

1,10 

1. 21 

1.20 

1.44 

i.3o 

1,69 

1, 4o 

1,96 

I.3o 

2,25 

l,6o 

2.56 

I.70 

2  89 

1,80 

3.24 

I.90 

36i 

2,00 

4,00 

Square  roots  of  the 

cubes  of  the  r  t  o« 

of  diameter,  or  v.il 

.       3 

ues  ol   02m 

Ratios  or 
d. 

l»o\ver  ? 

or  </2' 
i 

1.00 

1,000 

1.10 

1    1  54 

1.20 

1  3i5 

i.3o 

1.482 

1.40 
i.5o 

1.657 
1.S.37 

1,60 

2  024 

1,70 

2.217 

1.80 

2  4i5 

1,90 
,00 

2,6lQ 

2.828 

APPENDIX. 


525 


TABLE    X — continued. 

No.  5.  Tarred  Ropes. 

Stiffness  proportional  to  the  number  of  yarns. 

[These  ropes  are  usually  made  of  three  strands  twisted  around  each  other,  each  strand  being  com- 
d  of  a  certain  number  of  yarns,  also  twisted  about  each  other  in  the  same  manner. ! 


No.  of  yarns. 

Weight  of  1  foot  in 
length  of  rope. 

Natural  stiffness,  or 
value  of  K. 

Stiffness  for  loud   of 
1  lb.,  or  value  of  /. 

6 
i5 
3o 

lbs. 
0,021 X 
0,0497 
1,0137 

lbs. 
0,1 534 
0,7664 
2,6297 

lbs. 
0,0085198 
0,0198796 
0,0411799 

APPENDIX. 

No.  I. 

Take  the  usual  formulas  for  the  transformation  of  co-ordinates  from 
one    system  to    another,    both   being  rectangular,  viz : 

x  =  a  x'  +  b  y'  -f  c  z\ 

y  =  a'x'  +  b'y'  +  c'z',     I    .......  (l) 

z  =  a"x'+b"y'  +  c"z')\ 

in  which  a,  b,  <fec,  denote  the  cosines  of  the  angles  which  the  axes  of 

the    same    name    as   the   co-ordinates   into  which   they    are   respectively 

multiplied    make   with    the    axis   of  the    variable    in    the    first  member 

And   hence, 

xf  =  a  x  -f-  a'  y  -f-  a"  z, 

y'  =z  b  x  -f-  b'  y  +  b"  z,    I (2) 

z'  —  c  x  -f-  c'  y  +  c"  z  ; 

Multiply  the    first   of  (2)  by  b,  the   second   by  a,  and   take  the  dif- 
ference of  the  products ;  we  get 

bx'  -ay'  z=zy{a'b-ab')  +  z{a"b-ab")\  •  •  •  (3) 
again,  multiply  the  first  by  c,  the  third  by  a,  and  take  the  difference 
of  products ;    we    have 

ex'  —  az'  =  y(a'c-ac')  +  z{a"c-ac")         •     •     (4) 

Find  the    value    of  y  in  (4),  substitute    in   (3),  and  reduce,  we  find 

A  z  =  (b  c'  -  b'  c)  x'  +  (a'  c  -a  c')  y'  +  (ab'  -  a'  b)  z\ 


526 


APTENDIX. 


in   which 

A=.  c(a'bn  -a"b')+c'(a"b-  ab")  ■*-  c"  (ab'  -  a'  b), 

dividing    by   A,  and    subtracting    the    result  from  the  third    of  Eqs.  (1) 
we   have 

be'  -I 


ab'  —a'b\   f 

a    r  •  v  a    r  '  v         a7~)2  =  u' 

and   since    x't  y'  and  z'  are  wholly  arbitrary,  we  have 


(a" 


\    •  ,    (l„       a  c  —  ac  k  / ■  n 


a"  - 


bC'-b'C    =0;b"-a'C-aC'  =  0;C"-^-^  =  0;.(5) 

AAA 
transposing,    clearing   the    fraction,    squaring,    adding,    collecting    the    co- 
efficients of  c'2,  b'2,  a'2,  a*nd  reducing   by  the    relations 

a2  +  b2  +c2  =zl\a'2  +  b'2  +c'2zzl; 

a2  -f  b2  =  1  -  c2  ;  c2  -f  #2  =  1  —  a2;  a2  +  c2  =  1  -  b*. 

there   will   result 

i4L2=  1  -(««'  +  bb'  +  ccj. 
But 

aa'-f  6  6'  +  cc'  =  0, 

whence    il  =  1,  and,  Eqs.  (5), 

a"  =b(/  -  b'c;  b"  =za'c-ac'\  c"  ^  ab'  —  a' b. 

No.  II. 

To   find   the    radius   of  curvature   of  any    curve,  and    its   inclination 
to   the   co-ordinate   axes. 

Take    the    centre  of  curvature    as    the  centre  of  a  sphere  of  which 
the   radius   is   unity.      Through   the    same 
point   draw  the   line  O  X,  parallel   to   the 
axis  x,  and    another  O  T,    parallel  to    the 
tangent   to    the    arc    M  iV",  of    osculation. 
The    planes    of  these  lines  and  of   the  ra- 
dius of  curvature  will  cut  from   the  sphere 
the  spherical  triangle  A  B  C,  of  which  the 
side    B  C  is  90°,  A  C  the  angle  which  the  radius  of  curvature  makes 
with  the  axis   x,  and  A  B    the  angle  which  the  tangent  to    the    curve 
makes    with  the   same   axis.      Make 

p  =  O  R  =r  radius   of  curvature, 
&'=  AC;  c  =  AB;  C  =  A  C B. 


/9 


APPENDIX.  527 

Thel    will 

dx  a,  r       ■ 

cos  c '  =  -7-  =  sin  6  .  cos  C  ; 

a  s 
differentiating,   and   regarding   C  as  constant. 

d  x 
d  —  =  cos6'.  dB'.  cos  C; 
a  s 

but  d  &'.  cos  C    is    the    projection   of    the    arc    d  8'    on    the    oscillatory 
plane,    whence 

j  s,  ^        ds 

•  d8'.  cos  C  =  —  • 

P 
•Substituting  this  above,   we   find 

,  dx 
d  — 

Af  d  s 

cos  8 '  —  p  • : 

v       ds  ' 

and    denoting    by    6"    and    0"',   the     angles    which    the     radius    make* 
with  the   axes  y  and  z,    respectively,  we    may  write 

. dx  dy  , dz 

dT:  d~r  di~ 

m  CL  i>  ...  (IS  ....  u>  S  .    , 

cos  8'   =  p  •  — r-  ;       cos  6"  =  p-  — —  ;       cosd      =  p —  •  •     •   (1) 

ds  as  Is  v   ' 

Squaring,  adding   and   reducing   by    the   relation, 

cos2  8'  +  cos2  6"  +  cos2  8"'  =  1, 
we    have 

performing    the    operations    indicated   under    the    radical   sign,   and  redu- 
cing  by   the   relations 

d  s2  =  d  x2  4-  d  y2  -f-  d  z2, 

d2  sd  s  as  d2  x  dx  -f  d2  y  d  y  +  d2  z  d  z, 
we    find 

P  ~~  V(^  z)2  +  (d2y)2  +"(#  z)2  -  (aP 'is)2'  *  *  '  *  (2) 
If  5  be  taken  as  the  independent  variable,  then  will  ^5  =  0,  and 
Eqs.    (1)    and    (2)    become 

Af  d2  x  At,  d2  y  ....  d2  z  ... 

cos  &'    =  p  •  — -  ;   cos  8"  =  0  •  -5-f  ;    cosd"'  7=  p  ■  ^—  ;   •     •     (3) 
r    ds2  r    ds2  ds1 

ds2 
p  =       .      -  ;    .  .      .      .      .    (4) 

^/{d2  x)2  4-  (d2  y)2  +  id2  z)2  V   ' 


528  APPENDIX. 

No.  1 1  I 
To   integrate   the   partial    differential   equation 

•^    dq  da 

transpose   and   divide   by   Z>,    and   we   have 

dD~         7'D'dp' 
and   because   q  is   a   function   of  p   and    D,    we   hav& 

da      , ^      da     , 

j 

and   substituting   the   value   of  -7-=, 

a  J  J 

.  dq   D-dp —  y'V'dD 

di  =  rP 3 ; 


multiplying  and   dividing   by   y  •  D  •  p7         , 

1   --'        - 

D'-'p7        'dp — p7-dD 


dq  =  *l.ll9__      -1 


but 


dpi  £>* 

p7 

m 

D  —  p7        -dp—p7'dD  M  ry 


D2 


and   making 


1 


dp 

1 

1 

,:»/' 

• 

p7 

we 

may 

write 

1 

1 

dq: 

=  Ff 

(£>•< 

<S) 

and 

by 

integrating 

9  = 

■>(©■ 

> 

in 

which 

F,   denotes 

any 

arh 

itrary    function. 

APPENDIX.  529 


No.  IV. 


To  integrate  Equation  (414)'  of  the  text,  add  t«>  both  members 

d*r  (p 
dt.dr' 
and  we  have 

1       ,fdiy          d  r  y~i         a       ,rdra>  dry~» 
—  .d\  — —  +  a  -j—  I  =  —  .  d  I  -— "  +  a  — -?  I ; 

rf*         Vdt              d  r  A       dr        Ldt  dr  V 


and  making 


dry  d  r<p       T_ 


dt  dr 

the  above  may  be  written. 

dV  dV 


=  a  . 


</<   ~      '  dr  ' 
and  F  being  a  function  of  r  and  /,  we  have 

rrr        dV      j  dV     , 

d  V =  —  -dr  4-  — —  .dt; 
dr  dt  ' 

or.  bv  substitution  for  -r—   its  value  above, 

of/ 

rf  V  d  V 

dY=-—.(dr+adt)  =  -—.d(r  +  at)J 
dr  .  dr 

and  by  integration, 

dry  dry        TV  . 

dt  dr 

in  which  F'   is  any  arbitrary  function.     In  like  manner,  by  subtracting. 

d?  r  y 

a  . . 

dr.dt' 

from  both  members  of  Equation  (414)'.  we  find 

dry  dry 

K  = -77 -"•  77  ='  <'-•">> 

in  which  /'  denotes  any  arbitrary  function.     Whence,  by  addition, 


1 


630  AFPExm* 

»u4  by  subtraction, 


But 


Whence, 


■  dr<p      • )       (ire     , 

rf  r  9  ==  -r—  •  a  <  4-  — —  ,  d  r, 
at  dr 


dr^=~     J    (r+at).d(r  +  at)-  ~  .  f  .  (r- a  t)  d  (r  -  a  t)  , 
J.  CI  &  c» 

•ml,  by  integration, 

in  which  F  an  1  /  denote  the  primitive  functions  of  which  F  and  /'  are 
ube  derived. 


>r 


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